The fact is that there literally aren't enough places able to manufacture Warships. Most nations don't have more than one shipyard able to make KF cores at all, much less the compact cores used in Warships. It doesn't matter if you had one qualified crew or a million, building ships is the bottleneck.
I wouldn't say only 1.
Maybe the CC has only 1, IDR
The LC had 2 KF yards, 1 JS + 1 JS/WS
The FWL, FS, and DC were all looking at 3+ yards for JS/WS.
That said its still a small # of WS as you said & the JS & DS duties of a WS crewman probably don't overlap a ton for EVERY crewman.
I think early WS crews were stripped from JS & DS both & given some early cross training & then stuck on board a prototype WS in 3058 to start doing dry runs with it.
In the end only the FWL Navy comes close to being able to put much of strain on crew levels when they can always just pull more from the JS/DS fleets in small #s to fill out a crew or 2 given the small # of total ships being produced.