What's wrong with a username/password combo (besides all the usual things) or, if you want to get a bit more sophisticated, with having the user generate a private key for himself?

In addition to the usual problems, which are pretty serious to start with, you're relying on the client. To borrow from information security, the client is in the hands of the enemy. Sockpuppet (sybil in trust networks) attacks, where entity pretends to be many different users (aka sockpuppets), and impersonation attacks, where a user pretends to be someone they are not, are both well-documented and exceptionally common. Every forum package I can find relies on social taboos or simply ignoring the problem, followed by direct human administrator intervention, and most don't even make administrator intervention easy.

There are also very few sites that have integrated support for private-key-like technologies, and most forum packages are not readily compatible with even all password managers.

This isn't a problem that can be perfectly solved, true. But right now it's not even got bandaids.

Once again, with feeling :-D -- to which purpose? Generally speaking, if you run a forum all you need is a way to filter out idiots and trolls. Your regular users will figure out reputation on their own and their conclusions will be all different.

"Normal" social reputation runs into pretty significant issues as soon as your group size exceeds even fairly small groups -- I can imagine folk who could handle a couple thousand names, but it's common for a site to have orders of magnitude more users. These systems can provide useful tools for noticing and handling matters that are much more evident in pure data than in "expert judgments". But these are relatively minor benefits.

At a deeper level, a well-formed reputation system should encourage 'good' posting (posting that matches the expressed desires of the forum community) and discourage 'bad' posts (posting that goes against the expressed desires of the forum community), as well as reduce incentives toward me-too or this-is-wrong-stop responses.

This isn't without trade-offs : you'll implicitly make the forum's culture drift more slowly, and encourage surviving dissenters to be contrarians for whom the reputation system doesn't matter. But the existing reputation systems don't let you make that trade-off, and instead you have to decide whether to use a far more naive system that is very vulnerable to attack.

You can build an automated system to suit your fancy, but there's no guarantee (and, actually, a pretty solid bet) that it won't suit other people well.

To some extent -- spell-check and capitalization expectations for a writing community will be different than that of a video game or chemistry forum, help forums will expect shorter-lifespan users than the median community -- but a sizable number of these aspects are common to nearly all communities.

Why would Twitter or FB bother assigning reputation to users? They want to filter out bad actors and maximize their eyeballs and their revenue which generally means keeping users sufficiently happy and well-measured.

They have incentives toward keeping users. "Bad" posters are tautologically a disincentive for most users (exceptions: some folk do show revealed preferences for hearing from terrible people).