Mutations are random. What is not random is how human society selects from the pool of random mutations.
I am thankful that we do not have a global ground war in progress right now. I believe that political pressure to keep the healthy isolated on the front lines, with the sick going home to society -- this reverse version of containment -- was 1918 society selecting the more lethal mutations for increased reproduction.
From this standpoint, the herd immunity and flatten the curve approaches are no different. Both put severe versions of the mutated virus in an a hospital, which is an environment where eventually most hosts will have the virus, effectively deselecting virulence.
That said, not overwhelming hospitals probably means having more healthy doctors available for the virus to infect-- a slight selective pressure for virulence. This would be in contrast to herd immunity -- letting both a massive spike of people die along with doctors and then turning hospitals into simple pre-death, low-R_0 isolation wards.
The issue there though is that overwhelming hospitals via the herd immunity approach would decrease our ability to learn about the virus scientifically. Our ability to learn about pathogens and modify our behavior accordingly is probably our best hope for survival as a species, I think, so perhaps we should not overwhelm the hospitals with the herd immunity approach.