Some magic bullets from the past:
plenty of other vitamins for plenty of other diseases
clean water for preventing bacterial infections
washing hands to avoid massive childbirth mortality in hospitals
bed nets for malaria
If you look how difficult it was to figure out those, and/or how difficult it was to do such tiny changes even while knowledge of it became widespread, I don’t find possibility of trivial solution to willpower problem unlikely at all by outside view.
You cannot assume any α, and choose p based on it, for α depends on p. You just introduced a time loop into your example.