Different species, you can't generalize like that. It's pretty unclear what actually happened with H1N1. Scientists were able to resurrect the more virulent strain in the lab two decades ago and it was just as potent in lab animals...
Two possibilities are that it did in fact mutate to become "milder" or those strains were already circulating. Either way, H1N1 killed so quickly it ran out of victims and the highly lethal strain went extinct. Another notable aspect of H1N1 is that is mostly didn't kill directly, it made victims weaker to opportunistic lung infections and that's what killed them. Antibiotics have made this kind of attack vector much more difficult for viruses.
Omicron is only loosely analogous to the "flu fairy tale" as the major threat is Long COVID now and it is circulating at high levels. Other viruses have had vastly different natural histories, 1918 is only a single reference point, and a muddy one at that.
After reading this, I am less sure of the claim "most", but it seems that opportunistic infection was an important factor. There were a few unpleasant ways to die...
Two possibilities are that it did in fact mutate to become "milder" or those strains were already circulating. Either way, H1N1 killed so quickly it ran out of victims and the highly lethal strain went extinct. Another notable aspect of H1N1 is that is mostly didn't kill directly, it made victims weaker to opportunistic lung infections and that's what killed them. Antibiotics have made this kind of attack vector much more difficult for viruses.
Omicron is only loosely analogous to the "flu fairy tale" as the major threat is Long COVID now and it is circulating at high levels. Other viruses have had vastly different natural histories, 1918 is only a single reference point, and a muddy one at that.