Covid related links have saturated any searches on viruses in general, and the doom&gloom "reporting"is at the top. There are a lot of papers and examples of what I'm talking about, though, it's just irritating to get at.
Gain of function research gives us rapid and repeatable demonstrations of vital evolution. Viruses that don't kill their hosts and develop more and better mechanisms of infection or evasion of immune system defenses outcompete other variants. There are a lot of examples in animal viruses, but Sars and hiv are examples of recent relevance.
To be clear, I'm not making the claim that this happens to all viruses - evolution doesn't work like that. A virus that leaves you mostly functional for a few months before it kills you is not at all an unlikely scenario. Viruses also exist concurrent with other variants, and they can mutate rapidly. There's no hard and fast rule, just influences and constraints that can manifest as trends.
In my opinion, the ideal outcome for sars-cov2 at this point is that a super mild variant evolves that will confer robust natural immunity while spreading fast enough to prevent the spread of other variants. It could become more deadly - I hope it won't, and what we know of virus evolutionary pressures in humans hints that we could get lucky.
SARS and HiV are certainly not examples where a human virus has evolved to become more contagious but less deadly. SARS was not contagious enough to escape our control efforts, and HiV is mitigated by our anti viral drugs.
Indeed it would be nice if a less severe strain out competed current COVID, but such a thing has never happened before (as far as we know) - and as you point out there is no selection pressure for mildness when death/disability occurs some weeks after the infection has been cleared.
There are multiple strains of hiv. One of the papers I linked to is a detailed study investigating a less lethal, more infectious strain. Sars-cov1 also evolved a few strains that were more infectious and less lethal. Both effects, by the way, are usually associated with separate individual mutations. It's rare that a single change in DNA results in both effects.
Feel free to provide an example virus if you think I'm wrong!