There are, what, 6? 8? 12? 4T's described in S & H, and only the last two involved a president who could be argued to be a GC.
We could, of course, make that assumption. That is, now that the U.S. exists and has a president, we could assume that a 4T will always involve a GC president.
However, making that assumption, to me, makes the whole theory too deterministic, (as does the assumption that every turning must have a catalyst, which, I believe, S & H argue is not strictly the case). The S & H theory deals in statistical trends of entire generations, not determinative predictions of what will happen.
It's the difference between saying it will snow in Wisconsin next winter and saying it will snow next December 14 in Milwaukee.
The GC is a generational archetype, not one of a subset of all U.S. presidents.