David,

At first sight I think you might be right about this. Personally, I try to avoid using Weyl's inequality in this form, but rather some statement of the following form: If $|S_N| \geq \epsilon N$, and if $\epsilon > N^{-c}$, then there is some $q \leq \epsilon^{-C}$ such that the fractional part of $q\theta$ is at most $\epsilon^{-C}/N^d$.

In other words: if the exponential sum is large then $\theta$ is very close to a rational with denominator $q$.

You can prove this variant in much the same way, that is to say by repeated squaring. I don't think the inequality is typically stated in this way because the form you stated leads to better bounds in Waring's problem, where the factor of $(\log N)^C$ isn't very important. However, as you point out, it does seem to be important when talking about the equidistribution result.