This is commonly repeated but not really true, see http://www.iayork.com/MysteryRays/2007/08/26/rabbits-1-virus...

Switching from humans to human-infecting bacteria would be a very unlikely species jump.

I mean, it might not survive the longest if it's more infective, but evolution doesn't work like that, and we're worried about short term impacts. Yeah if it's really bad we'll eventually make it go extinct just like we made smallpox go extinct, but smallpox didn't adapt and survive and become less wild. We are the evolutionary pressure driving it to exinction - the virus right now is plenty happy surviving as the more virulent strain it is. It's survival of the good enough, not survival of the fittest, and a deadly strain is good enough to keep surviving.

I'm not sure that your first sentence is true. The symptomatic effect that a not-mild virus strain has can help it spread, for obvious reasons. It seems like there's an optimization problem there, with a sweet spot where the virus doesn't kill its host, but makes them cough and sneeze to spread to other hosts.

This is sort of true, in a Darwinian sense. But it's true because viruses which kill their hosts inhibit their own propagation! It doesn't help us at all if our goal is to prevent those people from being killed in the first place.