I don't think I follow the argument about presidents vs. prime ministers. Empirically, the UK does have a strong two-party system, so to the extent that they're

I don't think I follow the argument about presidents vs. prime ministers. Empirically, the UK does have a strong two-party system, so to the extent that they're a comparable case (i.e. setting aside minor things like the history leading to SNP and Sinn Fein), that's a point against your argument. In the US, we do in fact have parties that are dramatically different across state lines at the state level. This is changing a bit in the last few years, but it used to be only at the national level that the parties were internally homogenous. And even in Congress, except on the hot-button issues, there is some diversity across delegations from different states. Elizabeth Warren and Doug Jones are not the same. The problem is, if you go to a prime minister (e.g., if Speaker of the House got the powers of the President), then the individual state elections effectively become the national election. Imagine that there are no parties, but you and your friends have a few billion dollars to throw around, and want to make sure that the Prime Speaker is Nancy Pelosi. You'll do that by supporting politicians who would vote for her (donating, but also organizing get-out-the-vote drives, etc. - campaign finance laws can't prevent this). If anyone new wants to enter politics, they'll want to take advantage of your machine, which means that if they're on the fence about a policy they'll default to agreeing with you to make sure you pick them, and oops now you've reinvented parties. It's just an inevitable consequence of two facts put together: (1) FPTP voting, and (2) campaign organization having economies of scale, both in terms of money and in terms of voter attention span.