The problem with your analysis is that "go for the highest value" is not equivalent to "maximize expected value of the number you end up with"

For example, consider a similar game where you draw and then optionally re-draw (like in the original game) but the amount of money you win if you end up with a number p is p if p < 0.8 and p + b if p > 0.8, where b is some constant. Then you're also trying to "go for the highest value". However it is clear that in the limit of large b, your best strategy is to redraw if p < 0.8.