"Since the deadliest virus mutations kill their hosts, weaker virus mutations can spread (without killing their hosts) more quickly."
The problem with the SARS-COV2 virus is that infected individuals are infected before they're symptomatic, let alone dead. Thus there is no evolutionary pressure that would make this virus less lethal, as is the case for viruses that kill their hosts early in the course of infection.
What are some good references on asymptomatic transmission?
Early 2020 reports of asymptomatic transmission from South Korea were later retracted. There was a contact tracing study of 10 million people in Wuhan which found no asymptomatic transmission.
Unfortunately I think the media has really mixed up the words "presymptomatic" and "asymptomatic". You're right there is little evidence that people who never develop symptoms can spread COVID. But it seems likely that COVID can be spread up to 48 hours before symptoms develop, which still presents roughly the same problem.
I also anecdotally know a few different people who caught COVID from someone who felt fine at the time they interacted, so I'm quite confident it is possible, although I'm not sure I'd make a strong statement about the prevalence.
I believe that "asymptomatic" refers to an infected individual who will never get symptoms. "Presymptomatic" means the individual has no symptoms, but will develop them later.
As far as presymptomatic transmission, this is a good case study from Germany: https://wwwnc.cdc.gov/eid/article/27/4/20-4576_article. A 59 case cluster in which little/no asymptomatic transmission was observed, but 75% of transmission was presymptomatic.
Another interesting paper, also out of Germany covering 400k individuals, 25k infections: https://science.sciencemag.org/content/373/6551/eabi5273. I can't write a summary better than the one in the paper:
The role that individuals with asymptomatic or mildly symptomatic severe acute respiratory syndrome coronavirus 2 have in transmission of the virus is not well understood. Jones et al. investigated viral load in patients, comparing those showing few, if any, symptoms with hospitalized cases. Approximately 400,000 individuals, mostly from Berlin, were tested from February 2020 to March 2021 and about 6% tested positive. Of the 25,381 positive subjects, about 8% showed very high viral loads. People became infectious within 2 days of infection, and in hospitalized individuals, about 4 days elapsed from the start of virus shedding to the time of peak viral load, which occurred 1 to 3 days before the onset of symptoms. Overall, viral load was highly variable, but was about 10-fold higher in persons infected with the B.1.1.7 variant. Children had slightly lower viral loads than adults, although this difference may not be clinically significant.
The second paper is a bit dense, but there is a great discussion at https://www.youtube.com/watch?v=p4AR7qPxz_Q, with discussion of this paper beginning maybe 47:00. 1:14:08 beings a discussion about how a large number of cases might be caused by a small number of individuals. Interestingly, these individuals are not necessarily the same as those who go on to be seriously ill, hospitalized, or die.
In practice, the difference between presymptomatic and asymptomatic makes little difference. The key is that we have ample evidence that an infected person can infect others while they do not have symptoms.
The problem with the SARS-COV2 virus is that infected individuals are infected before they're symptomatic, let alone dead. Thus there is no evolutionary pressure that would make this virus less lethal, as is the case for viruses that kill their hosts early in the course of infection.