Some more countries use FPTP to decide part of their representatives, just not for all of them.

Then again, some countries are also democratic monarchies, with different heir picking rules.

The first what if you suggested is the one that will most likely bite you in the ass. If two candidates are similar and overwhelmingly popular, the more people that vote only for their most preferred are each making it more likely that the one candidate(s) they don't approve of have a more competitive number of votes. It's simply a bad strategy. While I think plenty will think that way in the shirt term, I would argue that the incentive is actually to not push that strategy, which will better serve everyone in the long term.

As for the lesser of two evils vote, that's not a problem, that's a feature. You can still vote for the lesser of two evils for the candidates you think has a chance AND for the candidates you actually approve of. The point of approval voting is to find the candidate that is MOST approved of. Even if angry single person voted for all of the candidates except their idea of the worst one, at the very least, that still eliminates the possibility of the least popular candidate from being elected. There can definitely be some spoilage there, an extremist candidate makes an easy target to rally against and a slightly less extreme though still fringe candidate gets more votes as a result, but it is still an unlikely outcome for them to have enough support to win a race that they are unpopular in.

For the person to be squeezed out, a majority of voters would need to have them as their second choice, but not meet the threshold of first choice votes to continue. They would be considered the preferred compromise candidate, but they get squeezed out of the race and polarizing the second round

In this context, not everyone's first choice, but the one that the majority could support against each other candidate individually.

For example if you have candidates A, B, and C, with 35%, 31%, and 34% of the 1st choice votes respectively. B gets eliminated. Only 50% of their voters put down a second choice, and their 2nd choice votes are almost evenly split between A and C, but slightly favor C. So lA and C end up with 49% and 51% of the remaining votes, respectively and C wins.

However, basically everyone that didn't put B as their first choice did put them as their second choice because they did NOT like their ideologically opposing candidate at all). So, if A hadn't run, the results would have been B with 65% of the vote and C with 35%. If C hadn't run, the results would have been B with 66% and A with 34%. Either way, B would have won by a landslide in those races, but instead B got eliminated in the first round which gave C the narrow win.

It seems like that maximum number of people would have at least approved of the outcome if B had won. Instead, only a minority of people people's votes ended up deciding the winner and everyone else sees the outcome as unfavorable.

This is I prefer approval voting. Two candidates never have to have split support and there is no worry about unsure elimination. So long as everyone honestly votes for whomever they would approve of, the candidate with the most support will win.