I'm willing to accept that in some designs, the first observation of p-value may be uniformly distributed over (0,1], but, as additional observations are made,

I'm willing to accept that in some designs, the first observation of p-value may be uniformly distributed over (0,1], but, as additional observations are made, the value should converge to 0 or 1. What would be the purpose, or usefulness of p-value being uniformly distributed if the null-hypothesis is true? It's much simpler to design things to converge to a single number. Edit: I have also considered that p-value could be uniformly distributed if the null-hypothesis is false (where you claimed true). I don't know the answer to that.