While those 3 factors are important you can't polish a turd and that's what a lot of the cheap drums come close to being AND the type of timber used for the shells does contribute to the tone and character of the sound. Guess what "music type stuff" I do when I'm not playing guitar or stuffing around in the studio............make drum shells, etc,.
.......the pic is a 5 ply Sapele snare I just finished setting up
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rpe,
Drums, like guitars can be a very personal thing and buying for someone else can be difficult. US made kits using US made Keller shells or Japanese made kits are usually at the top of the tree and for a number of companies drums from these countries are their upper level kits with their lower level kits being made in either China, Mexico or Indonesia which with the better known brands shouldn't be a problem as they normally have adequate QC, although some drums do come out of the same factories.
As a rough guide............:
Yamaha.......Japan and Indonesia
Tama..........Their own factories in Japan and China
Pearl...........Japan and Taiwan
Mapex.........Japan and China (Some Sonor models are made in the same Chinese factory)
Basix and Pacific........China and Mexico
D-Drum........China
Gretsch........Questionable but thought to be China (possibly some US made)
GMS............US and China
An alternative would be to speak to a custom builder in the US about making a kit using Keller shells, etc., as you will get good quality for less dollars..........if you're interested in this option, PM me and I'll give you some reliable guys to contact.