Yes, it makes sense. The hardest thing about recording with live instruments, especially drums, in a home environment is isolation (as you've already discovered) the sound of the room, and the quality of the recording gear. In a true studio environment, pretty much every drum would be individually miked, and have its own dedicated channel. Cymbals might share a mike, but kick, snare, toms & hi-hat would all be separate. They would also record in a room designed to record drums, so any room ambience would be complementary, though normally the rooms will be acoustically "dead" so that a very clean, neutral recording is made, which can then be treated in post-production. Finally, probably a few thousand dollars worth of mikes will be used, each one specific to its task. So that gives you an idea of how they get drums sounding good on albums. There are exceptions of course... if think there was an album recorded by the "Cowboy Junkies" where they reputedly used one mic for the whole band. Not sure if that's true though.

If at all possible, I'd have the drummer play to a click track, with the band, recording just the drums... that may not be possible with your setup. If not, then the scratch track idea should work.

After that, you'll want to record the bass and then rhythm guitars and any other rhythm instruments. From that point on, keep in mind that you're multi-tracking. There's no need to mix at this point... just keep recording additional tracks. You can pick and choose what you want to be able to hear... for instance, a vocalist may not need to hear a second guitar part, but just the bass, drums, and rhythm guitar.

Once you have everything recorded, you can do a final mix. Its still not a bad idea though that you have of doing mixes along the way so that way you can hear how things are fitting together. Sometimes when you record a whole bunch of tracks that you think will sound great, they actually don't mix well.

Do all this stuff just for fun, but before you put too much effort into a serious recording, consider perhaps booking time at a studio. A "semi-professional" studio can be really affordable, and having a set amount of time will force you to focus and get it done. Its also a really great experience to go and record with your band mates in a studio environment.