Drummers are in control of so much of the sound of a band. They really should be thinking like a producer. Create dynamics, keep the track evolving, create space, fill space. Behind the kit you can turn the whole band up and down. The rest of the band will (should) follow. A side-stick is very different to a hit on the centre of the snare, which is very different to a rimshot. These are your main "back beat" options. Your right hand can go from the rim of a drum, to the floor tom, closed hats, open hats, regularly opening hats (IE disco), ride bell, ride body, crashing the ride, riding the crash cymbals... This is how you either create space or fill it
As a producer, I want the song to remain interesting, so I want the beat, or at least the sound of the kit, to change from verse to chorus. If there are 3 verses, I want at least one of them to be significantly different to the others. Likewise if there's a bridge/middle 8 it should be different. I play in a band with no rhythm guitarist. A typical punk song for us would have closed hats in the verse, open hats in the chorus, ride bell in the bridge. A blues track would have closed hats in the verses, opening gradually for "builds", and I would play the ride cymbal to fill the space under solos
I think a lot of drummers just focus on what they're doing rather than listening to the mix as a whole, and it really shows.