Right. Grunge/metal drummers aren't really "right" for punk music. That style of drumming is too loose. It leaves too much empty space between beats. There's no tension or urgency in that style of drumming. "Punk drumming" is not always about playing fast, and it's not about filling every nook and cranny with fills. It is all about locking the bass, drums, and guitars together beat-for-beat on a song like yours. A basic way to do it is to lock the hats with the guitar downstrokes. Naturally there are BPM limits where a human being can no longer swing a stick on time repeatedly, but your song is nowhere near that limit. Programming your hats/ride to match the downpicking of the guitars is not out of the realm of possibility for a real punk drummer.