It is my understanding that McCarthy is struggling to get elected, because some more extreme Republicans regard him as 'too moderate'. Doesn't that mean that the choice is basically McCarthy or someone more extreme? In that case wouldn't it be sensible for the House Democrats to vote for McCarthy since, even if not really, he's still more in line with Democratic policies than the others?
"Never interrupt your enemy when he is making a mistake." (Attributed to Napoleon Bonaparte)
The Democrats, by refusing to assist the Republicans in electing a speaker of the house, highlight the disunity of the Republican Party. While the Republican Party is in disunity, they cannot enact any legislative efforts that the Democratic minority might object to, so this also achieves the goal of stopping Republican-supported bills from passing.
Additionally, there are no rules stating that the Speaker of the House has to be from the majority party. The Speaker traditionally is from the majority party because the majority party typically has enough votes to install them. So there is also a counter-question of whether it would be sensible for House Republicans to vote for Jeffries as a way to end the standoff, as he is currently the one closest to the required 218 votes. That is the outcome the Democrats want the most, and they are willing to wait and see if it happens.
McCarthy may be too moderate for some extremist Republicans, but he's not moderate in the classic sense, not in line with Democratic policies, and his anti-democratic antics may pose a threat not just to Democrats, but to democracy.
Looking at his history as minority leader: He supported far-right candidates such as Marjorie Taylor Greene and was a loyal Trump ally. He didn't accept the result of the 2020 election, voted against certifying the results, and spread false conspiracy theories about voter fraud. Incendiary remarks like his have been linked to the Jan 6 attack on the capitol and Democrats have called his actions "election subversion".
After what has been described as a "coup" failed, he started distancing himself a bit from the more extreme elements, but objected to an independent investigation into the attack.
His political positions are opposed to Democratic positions. He opposes abortions, environmentalism, doesn't accept the scientific consensus on climate change, opposes the affordable care act, and opposed DACA.
Putting the political differences aside, voting for someone who has a track record of rejecting and fighting to undermine democratic election results out of political considerations would be a danger not only to Democrats, but to democracy.
Given that there are other choices - such as hoping for moderate Republicans to vote for a Democratic speaker, for Republicans to put up a more moderate Republican speaker to win votes from the Democrats, or to go for an independent speaker and form a coalition government - it's not difficult to see why Democrats wouldn't jump at the chance to vote for McCarthy.
A number of Democrats have signaled that they are open to making a deal, but wouldn't support a Republican who objected to the 2020 election results.
Conservative commentator and never-Trumper David Frum, in "No Tears for Kevin McCarthy," (The Atlantic, 2023-01-05), argues that McCarthy is so compromised that Congress, the country and possibly even the Republicans are likely be better off with someone else, even if that other person is more ideologically conservative. And, indeed, that even with a speaker more ideologically conservative than McCarthy, the house may still end up with a less ideological majority.
Distaste for the anti-McCarthy faction, however, should not mislead anyone into supporting McCarthy. Very specifically, distaste for the anti-McCarthy faction should not mislead House Democrats into rescuing McCarthy. McCarthy has been frantically signaling for Democratic rescue. Speakers are elected by a majority of the representatives. If enough Democrats were to absent themselves, McCarthy could be elected speaker by the roughly 200 Republicans who do back him. But what’s the affirmative case for such rescue?
If McCarthy becomes speaker now, he will be a weak and precarious one—constantly at the mercy over the next two years of those 20-odd fringe Republicans voting against him this week. McCarthy will appease and accommodate them. When John Boehner was speaker, he dealt with the irreconcilable fringe by building majorities across the aisle. The 2015 budget deal that ended that year’s debt-ceiling crisis passed the House with only 174 Republican votes, augmented by 95 Democratic votes to reach the necessary 218 majority. But if McCarthy survives his present leadership test, he’ll do so only by committing never to repeat Boehner’s example. That commitment will have teeth, too, because McCarthy has reportedly agreed to allow any single disaffected Republican to call for a vote of confidence in his speakership if he displeases them. He has proposed to escape his immediate hostage crisis by handing himself over as a hostage forever.
That’s the beginning of the reason it would be better if he failed to win the speakership. If McCarthy somehow ekes out a win, he will be broken from the beginning—an officeholder who holds only the office, not the power of the office....
It would be better to have a speaker who can deliver than one who cannot, even if that speaker is more ideological than McCarthy. An ideologue can say “Yes” and have it mean something; a speaker who does not command a majority cannot.
By electing a more ideological speaker, Republicans may inadvertently shape a less ideological House majority. Imagine what this House will look like after a McCarthy defeat. Twenty Republican House members will have exposed 200 colleagues to national ridicule for reasons that even those 20 insurgents cannot coherently explain. Are the 200 now likely to follow the 20 into a fight to default on the U.S. debt? To slash American aid to Ukraine and hand the advantage to Russian President Vladimir Putin? To try to impeach President Joe Biden over some QAnon fantasy? To devote the next Congress to waging war on the FBI and other law-enforcement agencies? Or will they more likely say, “That’s enough from you—you have embarrassed us one time too many”?
TLDR: No, it would not be sensible for House Democrats to vote for McCarthy as house speaker. Even if he's personally "still more in line with Democratic policies than the others," he's likely to be compromised in a way that he will do what the Republican extremists ask in order to stay in power.
It is hard to say at this point in time because of various agendas that have been put forward for a Republican controlled house are things that Democrats don't want to happen. While it is true that whoever gets elected may go forward with those agenda items regardless there is no need for Democrats to help them happen sooner.
Those are both things that Democrats are not going to be too keen to get started.