Although I agree with you that the plurality voting system (aka FPTP) is a big problem, I think your theory of why it isn't changed is silly and overlooks the very real difficulties.
One of those difficulties is that there isn't a clear winner among the alternatives. Approval voting, the STV(/IRV), and the Borda Count all have adherents, who disparage the other two.
Worse, none of them is perfect. For example, STV and BC are both susceptible to strategic voting. I recall reading about some municipality that decided to try STV. IIRC, in the very next election they had two competent but extreme candidates at opposite ends of the political spectrum, and a centrist generally regarded as less competent (and who got a much smaller number of first-place votes than either of the others). The electorate was so evenly divided that the centrist won; the voters were so upset with this outcome that they ditched the STV and went back to PV (plurality). Evidently they would have been happier with one of the two extremists, even though that would have been the "wrong" choice for almost half of them.
We could object that none of the three candidates was obviously a great choice, and that if voters really preferred their opposite-extreme candidate over the centrist, they should have voted that way. But it's easy to see why they didn't. We could also object that AV would have given a better result in this case, picking whichever extreme candidate got the most votes and ignoring a widely-disliked centrist. But STV tends to be an easier sell, perhaps because it maintains the notion of "one person, one vote". And finally, we could object that they were too quick to pull the plug and should have given STV another chance, having learned a lesson about the importance of finding competent centrist candidates.
But their conclusion was, they tried an alternative system and it didn't work.
I tend to favor AV personally, as it's relatively resistant to pathological outcomes, but have to admit that it has another problem: it's the least tamper-resistant. With PV and STV, the total number of votes counted shouldn't be greater than the number of voters. AV has a much looser and less useful bound: the total number of votes shouldn't be greater than the product of the number of voters and the number of candidates. That leaves a lot more room for manipulation. In a world where our voting systems are already not as trustworthy as they should be, this is a problem.