I think using Bayesian regret misses a number of important things. 

It's somewhat unclear if it means utility in the sense of a function that maps preference relations to real numbers, or utility in axiological sense. If it's in the former sense then I think it misses a number of very important things. The first is that preferences are changed by the political process. The second is that people have stable preferences for terrible things like capital punishment. 

If it means it in the axiological sense then I don't think we have strong reason to believe that how people vote will be closely related and I think we have reason to believe it will be different systematically.  This also makes it vulnerable to some people having terrible outcomes.  

Lots of what I'm worried about with elected leaders are negative externalities. For instance, quite plausibly the main reasons Trump was bad was his opposition to climate change and rejecting democratic norms. The former harms mostly people in other countries and future generations, and the latter mostly future generations (and probably people in other countries too more than Americans although it's not obviously true.) 

It also doesn't account for dynamic affects of parties changing their platforms. My claim is that the overton window is real and important.  

I think that having strong political parties which the electoral system protects is good for stopping these things in rich democracies because I think the gatekeepers will systematically support the system that put them in power. I also think the set of polices the elite support is better in the axiological sense than those supported by the voting population. The catch here is that the US has weak political parties that are supported by electoral system. 

Coming back to this thread now having thought about it more. Speaking from my personal experience as an American citizen, I think the spoiler effect lowers voters' confidence in the electoral system, especially that of idealistic, young voters.

I was a Bernie supporter in 2016. I wasn't excited about Hillary being the Democratic nominee, but because I understood the incentive structure created by FPTP, I chose to support Hillary in the general election because I really hated Trump. But, there was a substantial number of Bernie supporters who defected from the Democratic base after Bernie lost. Mainstream Democrats seemed to be shaming them into voting for Hillary, on the grounds that:

• if you vote for Jill Stein (the Green candidate) instead of Hillary, Trump will win.
• if you vote for Gary Johnson (the Libertarian candidate) instead of Hillary, Trump will win.
• if you write in Bernie, Trump will win.
• if you don't vote, Trump will win.

This has been called vote-shaming, and I think it makes American political culture a lot more toxic because it pits ideologically similar people (like center-left and far-left progressives) against each other. Many people don't vote at all, both because of voter suppression, and because they don't feel represented by the major candidates. Eligible non-voters in 2016 were also more likely to be younger, less educated, less affluent, and non-White (source), which suggests that the system is not representing these groups as well as it could be. It is a problem that citizens of the world's oldest continuously running democracy feel disempowered - it means that the government is not as responsive to citizens' interests as it should be. Vote-shaming puts the blame on individuals for not voting, instead of the system for causing vote-splitting.

Just so you all don't think that this only happens on the left: I have a friend who didn't really like either major candidate. He leans conservative and strikes me as someone who might have preferred the Libertarian Party or Bernie Sanders. Despite not liking Trump that much, he voted for Trump in the 2016 general, because he thought Hillary was worse.

Some statistics:

• In 2016, just 54.8% of the voting-age population (VAP) voted in the presidential election; 59.2% of the voting-eligible population (VEP) voted.
• In 2020, this increased to 62% of the VAP and 66.7% of the VEP. (source)

I think that increasing voter turnout would make the government more responsive to citizens' interests, and I think changing the voting system we use would help with this because it would help citizens feel more empowered to vote.

Note: I'm not saying that vote-splitting, or even problems with the voting mechanism in general, is the only issue with the U.S. electoral system. I think there could be other problems introduced by a new voting system such as approval voting - practical problems that degrade the political system similarly to the way that I think vote-splitting does (since we know that no voting system is theoretically perfect).