No, it doesn't.
https://www.politifact.com/factchecks/2021/dec/08/facebook-p...
> The claim makes a broad generalization about pathogens that’s not supported by science. It has been well-documented that pathogens can evolve to be more virulent. And many viruses, including HIV and Ebola, have in fact become more lethal over time.
As you note, smallpox disproves the idea that viruses automatically become mild. It had thousands of years to do so and carried a 30% mortality rate still.
If everything survives, the resulting community can live with each other and that requires a sufficiently mild virus. However that doesn't help predict the future, because you can't enforce the precondition of everyone surviving.
So a more lethal virus during a pandemic must also be more transmissible, as it's actively competing for the same susceptible hosts with other circulating strains. If another strain gets there first, chances are it's going to prime that host's immune system sufficiently to prevent reinfection.
Which means it needs not just more lethal mutations, but also more transmissible ones. The probability of all of those coinciding seems low, as the former are more likely to arise in immunocompromised hosts, and the latter in circulation among dense populations, no?