But unfortunately, your last inequality is only a lower bound and not an upper bound. Let m be its right hand side. Then do we have $r \le floor(m)+1$?

Yes, indeed. The case I had in mind was $f(x) = (4x(1-x))^{n/2}/2$, such that $f(1/2) = 1/2$. That was where I observed the limit of 1/3. When I now do some experiments on different polynomials, a different limit is reached (e.g., for $f(x) = (4x(1-x))^{n/2}(3/4), f(1/2) = 3/4$, a limit of 5/4 appears to be reached).

@WillSawin: Yes, exactly.