Skipping your first question (I don't know of one) and going directly to
If not, why is Theresa May not using the threat of this outcome as leverage against the EU?
It doesn't work so well as a threat. Consider the following possible actions:
- Ireland leaves the Good Friday deal.
- The European Union accepts Northern Ireland as a continuing part of the EU.
These have at least as many negatives for the United Kingdom as for (the Republic of) Ireland.
TL;DR: May won't do it because the threat is too likely to boomerang on her, given her government's reliance on Northern Irish MPs.
Northern Ireland secession
The general argument against the second option is that countries like Spain might see it as a bad precedent. But in this case, it's no longer separatists in Northern Ireland driving things but the UK. If Spain becomes more worried about the UK than the Catalans, they might drop their opposition and allow Northern Ireland to leave the UK and stay in the EU. They could describe this as the UK leaving Northern Ireland rather than Northern Ireland leaving the UK. I.e. it's the UK that is acting in bad faith, not the Northern Irish.
This does not require Ireland to annex Northern Ireland. Northern Ireland could simply secede from the UK. If the EU then allows it to join as a new country, Northern Ireland would be mostly back to the status quo. It would then be harder to travel from Northern Ireland to the UK or vice versa (or trade), but Northern Ireland maintains the same relationship with Ireland and the EU. Some of the Northern Irish may prefer this result to staying in the UK and leaving the EU even without threats to the Good Friday deal.
If Northern Island were to secede, what could the UK do? Invade? Blockade? It would have to blockade all of Ireland, which would be unlikely to receive United Nations approval. It would be sorely short of allies in that situation. Meanwhile, Northern Ireland would have the support of the EU in general and Ireland in particular.
It's not clear to me why you think this scenario would threaten the EU's legitimacy. It would be clear to everyone that the UK was not negotiating in good faith. Northern Ireland is currently part of the EU and voted to stay in the EU. If it were to choose to stay in the EU rather than in the UK, that would attack the legitimacy of the UK more than the EU. The only reasons that hasn't been considered anyway is that the UK opposes it and parts of the EU see it as a bad precedent. But in this scenario, it is only when a country leaves the EU that parts of it can stay. Since Spain is not planning to leave the EU, this doesn't really apply to them.
Political ramifications of the end of the Good Friday deal
We also should consider who the biggest proponent of the Good Friday deal in the UK is. It's basically the Northern Irish. And on whom is Theresa May relying to add just enough votes to allow her to survive a "no confidence" vote? The Northern Irish. So this threat may make it more likely that she would stop being Prime Minister. While she would technically still be a Member of Parliament (MP), this would likely end her career in leadership. As such, this threat may hit her worse than anyone else.
Of course, perhaps that result would be best for the Conservative party. May would be out as Prime Minister. Labour would likely win the resulting election. Labour would be stuck trying to negotiate with the EU. Currently, the Conservatives are split between those for and against Brexit, while Labour is united against compromise. But if Labour had the government, then those against Brexit would be against Jeremy Corbyn, who is pro-Brexit. So it would be Labour that would be splitting, while the Conservatives can criticize from the sidelines.
It's even possible that neither Labour nor the Conservatives would be able to form a government. It might be that the Liberal Democrats and Scottish National Party take enough votes to prevent either side from forming a government without them and demand an end to Brexit as their price for joining a government. That might cause both Labour and the Conservatives to split into pro- and anti-Brexit factions. The UK might end up with a pro-Brexit or anti-Brexit government that will dissolve again as soon as Brexit is resolved (terminated, negotiated, or whatever).
I remain unconvinced that Chequers has any constituency. I think that most Chequers supporters are really anti-Brexit. That's why I think it would take two elections. In the first one, Labour paints over these problems by being not-May. But then they can't find a compromise that anyone likes. Second election has people realizing that the real choices are no-deal Brexit or staying in the EU. There is no compromise that the EU will make that is preferable to both those options for most people. Eventually I think that people have to choose between the available if extreme options.
The fundamental problem is that pro-Brexit has claimed benefits that are simply unobtainable. If you ask most people if they would prefer the Chequers deal to staying in the EU, it seems unlikely that they would choose Chequers. Chequers basically keeps the unpopular policies of the EU while giving up things like a vote in EU policy.
The problem here is that neither the Conservatives nor Labour are really pro-Brexit. So there are a bunch of people who would prefer no Brexit instead negotiating Brexit, thus Chequers. To come to a real agreement, they are going to have to choose Brexit or not. The only Brexit deal that a UK government can definitely deliver is no deal. Until they realize that and vote on what really matters to people, this is going to be a political turkey.
That's why I'm saying that both the Conservatives and Labour will have to split into new parties. Because currently, there is no one representing the anti-Brexit opinion in either the government or the opposition. Only small parties like the Liberal Democrats and SNP are openly anti-Brexit.
The hardline no-deal people seem to think that they would win another Brexit election. They may be right. But the current system means that no-deal Brexit is inevitable, because there is no way to get enough votes for an unpopular compromise. Labour opposes any deal, because they want a new election that they can win. The Liberal Democrats and SNP also want the chance to make gains in a new election. Some of the Conservatives prefer a new election, because they think that Chequers is too EU-friendly. How does a deal get a majority?
Many Conservatives are anti-Brexit but stuck supporting the referendum results. However, if they lost the government, then they could use that to go back to their natural preferences. Clearly if the referendum were supported, May would have won the election. If she doesn't win, then that means that the May position, negotiating any Brexit deal that retains the free trade agreement, is not supported.
There are basically three options:
- No deal Brexit.
- Cancel Brexit and stay in the EU.
- Accept EU regulation without any say in it.
I suspect that pretty much everyone prefers at least one, if not both, of the first two options to the third option. Some people may prefer the third option to the first option. But those people will almost always support the second option to either of the others. I don't believe that anyone really supports the third option over the second option. So the only real chance for the third option, a Chequers-style deal is that party loyalty holds enough Conservatives that enough Labour people can switch and vote it into effect. That seems unlikely.
The problem is that if Labour (or the Liberal Democrats or the SNP) politicians vote for a compromise Brexit, they are then vulnerable to someone running against the compromise who gets both the pro-Brexit and anti-Brexit vote in the next election. Perhaps they would survive, but it wouldn't seem likely going into the vote. All the feedback would be against the compromise.
The result of this may be to empower no-deal proponents. But it's hard to see how to counter that. Beyond that, it's not necessarily so. Many people who voted Leave expressed remorse later. Remain might win a second referendum. Remain might become even more popular if the choices were Chequers and Remain. Because Chequers gives neither side anything that it wants. It's not Remain and it doesn't give the promised benefits of Leave.
Similarly, Remain versus No Deal might drop on the side of Remain. Some of the people who voted Remain didn't think through what No Deal might involve, because Leave promised that the EU would negotiate. Now that it's clear that they won't, maybe people change their minds.
Of course, it is also possible that UK pro-Brexit politicians again convince people that Leave is better than Remain. After all, who want to stay in a union with a bunch of people who won't negotiate?
I don't think that it is clear who would win a new Leave/Remain election. I'm just saying that the Conservatives are currently losing. The question may not be if Corbyn will be Prime Minister, but when and how long. For Conservatives, sooner would be better. Because they want Labour to bear as much of the burden of governing as possible while offering them as few actual benefits as possible.
Even if May makes only an implicit threat, she could very well be forced by the Northern Irish MPs to then make an explicit promise to work out a deal that supports the Good Friday provisions. Because this will be really important to the Northern Irish and the opposition there is likely to use it to argue that they made a mistake when they agreed to prop up the May government in the first place. That's what political opponents do. They construe any statement as negatively as possible.
This is really the whole problem with Leave again. They took an option with the expectation of compromise. But what if the other side doesn't compromise? This is a bad bluff. And if it's not a bluff, the consequences are as negative to the UK and especially May as to the EU. She could easily end up with less leverage, as everyone would then know that she needed a deal on at least that one area.
Things might be different if the Good Friday deal was unpopular in Northern Ireland. Or if Brexit were popular in Northern Ireland. But that's not how it is. The Good Friday deal is very popular in Northern Ireland and not very important in Great Britain. And Northern Ireland voted against Brexit. It wouldn't take so much to split Northern Ireland from Great Britain.
This seems like something that May would be very careful to avoid even suggesting. It's not much of a threat to the EU except that it could lead to a situation where Ireland wasn't enforcing the customs union properly. It's far more of a threat to her. Because if she doesn't make the threat explicit, then it seems unlikely that the EU will take it seriously. But internal opponents may choose to take it seriously even if they don't believe that she will follow through. So she gets all the disadvantages of the explicit threat with none of the advantages.
If she can't even make the threat openly, how would she ever carry through on it?