You build up your skills on the snare and you transpose those skills to the entire kit.
The rudiments, timing, etc. When you play the kit you are playing the same things that you can play on a snare drum in different ways.
Learning rudiments and snare drum solos is great for getting your technique together. With just a snare drum you can work on touch, phrasing, etc. One's drum set ability will only be as good as what one can do on the snare drum...
Hands require extra attention in order to fully develop. You need to have nice clean relaxed singles and doubles and the snare/pad work will get you there. Every drummer with great hands has paid their dues on the pad.
First of all, snare drum solo stuff is a part of many songs in pretty much any musical style. Start playing around with subdivisions and accents and you can build your whole drum set style on that.
Take any rudiment and move your habds around the kit and make a groove or fill out of it.
Take eg. a paradiddle with the RH on the hi-hat and LF on the snare. That's a pretty cool grovve all it's own. Move accents around the toms or crash w/BD. OK, that's the door opener. Then do the same with all inversions and all accent possibilities.
Try the same thing with any rudiment and also replace hand stroke with the BD.
Rolls are cool on the hi-hat with accent on the edge or on a splash. How about between the hi-hat and ride?
It never ends.
Everything can also be done between any limbs.
Apart from the most basic of rock beats the rudiments are everywhere and the demands for clean and fast singles and doubles are proved in any song with just slightly involved drumming.
Good control and knowing the most economical strokes makes for a nice flow in all sorts of groove playing. Moving around the kit demands a more movement and there are many surfaces to get used to, but ultimately it should be just as comfortable as a working on the snare.
They're the source of all drumming.
If you practice diligently and get a bit creative the doors of possibilities should gradually open themselves up.