Since the long form may be too detailed, here a quick TL;DR:
- Keep it as it is now.
- The accepted should be pinned.
- Accepting an answer is (usually) a quality statement
- This is true for 'Problem' questions as Wizzwizz4 mentiones
- But as well for any other, as the OP can/must decide which answer really hits his question.
- Votes in contrast are quite often just likes for whatever reason.
- Likes are not a quality statement
- Changing that solves no problem and serves no purpose.
- Changing will only worsen the experience of (new) users looking for a solution/answer to the same issue.
The most pertinent one is that pinning exacerbates the Fastest Gun problem
Somehow it seems as if you're obsessed with this assumption, having it used several times for attempted accusations.
Bringing it up here seems rather frivolous as the mechanic of sorting accepted answers to the top is exactly what prevents any benefit of answering first (see). After all, it's the questions author acting as filter with (some) knowledge about the topic - even if it's only newly acquired one.
askers tend to hastily mark answers as accepted, often choosing the first answer they receive, which may only superficially seem good.
Accepting an answer is, in my experience, usually a rather slow process on RC.SE. Unless it's an obvious case - like a pointer to a missing source. It didn't happen often that I have seen an answer accepted early one. I noticed this only a few (very few) times with obvious first time users, making it in itself notable.
It's quite common that askers do wait several days (or for better answers) until they decide to mark one as accepted. Sure, this may be different on other sites, but RC.SE is not the average ask for code site.
Using Wizzwizz4's script with a 'history' tag shows that for many questions the accepted is not the first given. Roughly half picks a later one. Given, that script my benefit from more refinement and especially providing summary results, but it gives a good first indication about an assumption being worth to follow up.
And the ‘better’ answer doesn’t even have to be posted seconds after; the Fastest Gun can also fire in slow motion.
Fixation on FGITW won't get more useful by adding sub-theories.
Not to blow my own trumpet, but it does make me sad that my answer about Windows 3.1 dithering,
That question possess a general problem for complex technical issues - they can't be easy followed without deep knowledge or a considerable effort to validate the claims. So people may rather act by 'gut-vaue'.
which I posted five years after the misleading accepted answer,
That adds of course as well.
It’s probably the most egregious example of the FGITW problem,
Sorry, but arguing with an answer made five years later and then calling anything before FGITW is by no standard a serious contribution.
and how pinning makes the problem worse.
But it will go up to second. Just the example doesn't help advance the issue, as the post happened so late, that it might, even without an accepted answer, never catch up.
(And I guess we all can agree that 'last post first' would be the worst of all sorting orders, making late coming 'something' the top answer)
Not even my vote-weighing script makes it any better, not without making some questionable adjustments to the weighing factors.
I would say any vote weighting factor is questionable. Who tells that a down-vote is in any way more telling than an up-vote? There are quite numerous reasons for down-votes. May it be malicious, like for not liking the one answering, over cultural(*1), all the way to simply not liking the content given/expecting a different answer.
The cultural part is for example easy to see with answers that state facts which do not go well with fan based beliefs - e.g. pointing out that the C64 is just a pimped up game console (max machine). Such answer receive an above average down-votes.
One of my highest ranking answers is as well a nice example, here the OP accepts a different answer. Both were given within 12 hours. Still did my answer receive about 3 times as many votes (151 vs. 53). Looking thru my answers shows several similar examples. Notably more than one from the same asker with the same result, so being a nice example for an askers bias.
Also, when talking about this, we should not forget people who post questions and then play moving target with the explicit goal of invalidating existing Answers they may not like for either reason. This as well screws answer ranking.
Many more examples where better answers are available can be made - as well as other where askers simply don't accept an answer despite being obvious, but I guess flaws of over detailed argumentation will outwight adding more.
Long story Short:
While there are some questions better left without an 'accepted' answer, I do believe that accepting is a very valuable tool where an asker can use his (new found) judgement to mark out the one he found more useful,which helped him to understand the issue he asked about.
Never forget, this is a social media site - we are people with all their rough edges and weird ways to interact. Any too 'fine' regulation will create even more trouble than they solve.