Ah, very nice pictures and observations. I agree: that main snare looks like a supraphonic with the standard lugs, 5x14. The auxiliary snare I can't make it--definitely brass colored, but it too looks like a 5" depth. Hmmm.
Regarding the vintage kit, I don't know. He is using the Yamaha subkick device on the bass drum. However, take a look at this Yamaha kit from the late 1960s--could it be something like this? This kit has a similar finish with the rail mount--and after looking at your photo again, I definitely think ?uestlove may be using this Yamaha birch kit from the '60s: