You take what happened in the two elections previously (and I know technically we don't vote for PMs, but they drive the agenda of the party).

2015 we voted for Cameron, ended up with May then Johnson 2019 we voted for Johnson, ended up with Truss(!!) then Sunak(!)

There are efforts to make this happen in the us starting locally and working up. The states are left to decide how they implement elections on their own with a couple of exceptions. There is a tragedy of the commons aspect to it though, as if some states adopt proportional representation but not others the ones that do not adopt it gain advantage. Ranked choice voting is taking hold much faster than pr in the us, and it is pretty slow too. It can happen though. Both are viewed as being left leaning, which doesn't really make sense to me.

Yep and the coalitions are famous for exemplifying the concept of "poldering:" https://en.wikipedia.org/wiki/Polder_model

If their minor party doesn't end up as part of the governing coalition, there's no sense in which people feel like their vote wound up having no effect?

That sounds like kind of a mirror of some of peoples biggest complaints regarding bureaucracy and committees. Deadlock can not only be worse than an imperfect solution, it can be weaponized by a minority to exert outsized power and extract otherwise unthinkable concessions. We see this sometimes in the US House, where more fringe or radical groups within parties can block the literally functioning of the actual country, safe in their assumptions that the two parties will not form a majority coalition and that the parties as a whole will take more damage from the fallout than the radical groups.

I'm not saying that that makes the system worse, mind you. I'm not even saying you're wrong that it's a better system. I just think anyone who thinks any one system is the easy, obvious fix to fair and just representational government is either shortsighted, or has different priorities than I do.

My view is it's always organised elites making the decisions, no matter the system. Nominally left-wing parties often make brazen right-wing moves, and vice-versa. The votes that matter are those of the MPs, Congress members etc. which are always influenced by a range of factors and organised factions. That's the actual decision-making mechanism.