If the UK left the EU without a trade deal, trade with the EU would continue because it would be illegal to offer the UK worse trade terms than any other third country under WTO most-favoured nation rules. Is this correct?
That is correct, the UK will have the same terms that any other non-EU country1.
Member of the World Trade Organization grant the tariffs of MFNs to the other members of the WTO. The UK is a member of the WTO independently of the EU2.
That also means that the EU cannot simply give the UK a better trade deal on its own, because then the other WTO countries would also benefit from it. It also works the other way; the UK cannot privilege just the EU (on account of the Ulster-EIRE border, for example).
It does not include financial services, which are an important part of the UK economy. Moreover, some of EUs Free Trade Agreeements include MFN clauses affecting financial services, so opening up to the UK would also open to those countries.
1 Custom Unions are exempt from the MFN rules; the fact that there are no tariffs between France and Germany does not mean that Germany's MFNs will have no tariffs.
2 The EU as itself is a member of the WTO, but all of its countries are also members on their own.
Trade could certainly continue under the basic WTO rules. It wouldn't continue under the free trade rules of the single market. Quoting from a WTO webpage explaining the concept of "most-favoured-nation":
Some exceptions are allowed. For example, countries can set up a free trade agreement that applies only to goods traded within the group — discriminating against goods from outside. Or they can give developing countries special access to their markets. Or a country can raise barriers against products that are considered to be traded unfairly from specific countries. And in services, countries are allowed, in limited circumstances, to discriminate.
The European Single Market is an example of one such group.
Of course, from the point of view of EU-UK trade, firstly the WTO tariff rules are still a fairly stiff trade barrier (and both sides have to apply them, to preserve the most-favoured nation status rules), secondly there is limited coverage of services, which makes up a sizeable proportion of the UK economy.