Hmm... if the drummer you play with can't keep time, then you won't find the groove no matter how hard you look.
"The groove" and "time" aren't exactly the same thing. Some grooves require you to play in front of or behind the beat. And if you try to play straight 4/4 over something with a swing-funk feel, you'll wonder why you're not grooving along. And if you play a straight 1 when it's a 1 with a push, you'll wonder why you keep missing the start of the note even though you're playing bang on 1.
If you just can't keep time, that's something else. Practice with a metronome - eventually you'll get there.
Tapping your foot just doesn't come into it. Drummers don't tap their feet. I've seen a fair few musicians play completely in time together but have their feet tapping hideously out of sync. It also really annoys me when I see singers tap their hand against a microphone "in time" with the beat. It looks daft - especially 'cos they're always out.