I'm not the commenter above, but I appreciate your polite and good-faith discussion of this issue, so I hope you don't mind me presenting the counter-argument in their absence.

> If they could do this they'd form a coalition and form a government, but they don't.

In order to form a coalition, parties need to collectively have enough elected MPs to hold a majority in the House of Commons. A major point we are making is that the electoral system in the UK allows the Conservatives to gain a disproportionate number of seats, unreasonably preventing the other parties from being in a position to form such a coalition.

As the House of Commons Library itself explains[0]:

"In 2019 the Conservatives got one seat for every 38,264 votes, while Labour got one seat for every 50,837 votes. It took many more votes to elect a Lib Dem (336,038) and Green MP (866,435), but far fewer to elect an SNP MP (25,883)."

(Note that the fact that the SNP are over-represented in Westminster is little comfort, given that they only field candidates in a minority of Commons constituencies, and I have covered the strength of their support in the Scottish parliament elections, with its different voting system, in another comment. Also, the SNP are principled enough to oppose FPTP despite the fact that they benefit from it in Westminster[1]).

To steelman your position a little, perhaps by "form a coalition" you mean some sort of electoral pact before an election. I mentioned in another comment that this has been tried in a few cases, but there are severe political problems with the idea of parties telling their candidates not to stand in specific constituencies, hoping that their voters will still bother to turn out and support the "second best" alternative. Not only is there no guarantee, under FPTP, that this strategy would work, but it would start to make a mockery of the idea of having political parties at all if some of them are actively avoiding standing candidates.

> You can look at them and call many of them 'left', but other than that they're not really compatible.

It's true that there are as many policies which divide these parties as unite them, but it's more accurate to say that a Principal Component Analysis would detect what we call the "political spectrum" of left and right wing parties as the most significant dimension of political opinion.

Just because you would need millions of political parties to fully capture all the individual political positions of every voter doesn't mean that there aren't broad trends, and in particular it is possible that a (weighted) random policy picked from a non-Conservative party could be on average more popular with the electorate than the equivalent Conservative party policy.

Ideally I would express this as candidates, parties, and voters in an n-dimensional space, so we could talk about drawing hyperplanes to define parliamentary majorities, but that's probably too abstract to be helpful. If you haven't played around with Nicky Case's "To Build A Better Ballot"[2], which helps visualise something similar in two dimensions, I highly recommend it.

> For example are the Liberal Democrats part of your left block that would win? But when given the choice they coalesce more readily with the Conservatives in practice.

We really only have one data point for how readily the Liberal Democrats form coalitions with different parties, and the specific political environment in 2010 is really not helpful to extrapolate that point from. Not only would Labour and the Liberal Democrats not have had enough MPs to hold a majority at the time, but the reputational damage of the coalition with the Conservatives has no doubt changed the political calculus that the Liberal Democrats would apply if a similar situation were to happen again.

[0] https://commonslibrary.parliament.uk/general-election-2019-t...

[1] https://www.independent.co.uk/news/uk/politics/snp-would-vot...

[2] https://ncase.me/ballot/