I'll give you an example. For normal distrubution N(mu,sigma^2), 68% of data will fall into the range
(mu+/-sigma) whose width is 2 sigma. For a uniform distribution (mu+/-delta). 68% of data will fall
into (mu+/-0.68delta) whose width is 2*delta. Find the
delta so that the standard deviation of this unioform
distribution will be the same as sigma. Then check
which width is bigger 2*sigma or 2*delta. I believe
2*sigma is smaller which means it is easier to predict
where mu is for the normal distribiution.

If this counter exsample holds, then your conclusion
is wrong!