Does it sound like crap on it's own in the room, or like crap on the recording, or both? There are big differences.
If you don't like the way it sounds live, in the room, then tune it differently or use a different snare. Try different heads. Keep in mind though that the drummer is a member of the band too, and it's ultimately his sound. If he doesn't tell you how to play guitar or bass or sing, leave him alone about his snare. Tight snares play and feel differently than looser snares and maybe that's the feel he likes. If you think you can change his mind, give it a try. I'm personally open to drum suggestions from my bandmates, but that doesn't mean I follow them. You gotta be careful here.
If you like the way it sounds when you play live, but not recorded, then it's your miking technique and/or room and/or equipment. It's tricky capturing the true sound of anything, nevermind a banging snare drum. Keep experimenting with different mics, if you have them, and mic placement. Snares like EQ and compression. And keep in mind that you listen to thrash metal and are looking for "that" snare sound. There's not much natural happening with drum tracks from that genre. Sample replacement very likely in everything you listen to. That sound you're trying to emulate is probably a sample recorded in ideal conditions with ideal mics with and ideal drum. Do it if you want to and your drummer doesn't mind..
But, that being said, for the good of the band, you can all work together to get a great sounding band track... including the snare drum! If you want a deeper snare, why don't you try using a larger snare! Maybe a 10" or 12" deep instead of a 5" or 7" deep. This will give you more body and pitch depth and it will still have a nice tension for your drummer to play. I personally rarely use a bottom snare mic, but for a deeper snare, you will need to to get the snare buzz, therefore I always use one on a snare that is more than 8" deep. There is also something else you will need to address... and that is phase cancellation between all the mics. Mics that are out of phase can kill your recorded sound! If you are using two mics on the snare... a top primary mic and a bottom mic for the 'buzz', the you should definitely check the phase correlation between the top and bottom mic snare mics. I find that 90% of the time, it is out of phase with the top mic. Also, check the phase of all the mics on the kit to see if they are 'out of phase' with one another. If they are, they will actually "remove" sound instead of "add" sound to the recording. You can do this by listening in mono (or panning all the channel playbacks to one side...) and 'flipping' the phase on each individual drum channel. If you have no phase flop on your console, then listen to what happens when you start adding mics one by one to the mix. Does the sound diminish or is it "adding"? Phase problems are caused by bad mic cables, incorrectly positioned mics, and a number of other things. Most Radio Shack stores will have "phase flop" barrels for sale. I got mine at Pacific radio in L.A. You can also flop the phase with a plug-in at the mixing stage, most plug-ins offer a phase reversal option. You should check out my book, "Drum Recording Handbook" published by Hal Leonard Publishing to learn more about the art of drum recording. Anyway, good luck and have fun!! 
