Gear does matter, a lot of key elements to thrash are the amp, speaker, boost and a parametric or graphic EQ in the loop (preferably switchable to not mess with lead tones, as that rhythm notch tends to lead to meh single note tones). Don't have to use the EQ for the mid cut, either. Another big trick for getting tight rhythm sounds is to cut bass by using a tube screamer or other boost with a bit of a bass cut, treble booster (Judas Priest), or EQ, then bring the bass back up after the preamp, so you have big, but not loose & distorted low end.
But if your pickup isn't feeding the amp the frequencies you want to distort, that's tough (and usually noisy) to get back. Midscooped pickups can be far worse (broader cut to mids) than outboard EQ (though rarely as bad as an amp with the mids on zero...)
But there are elements of attack and feel that pickups influence heavily. Sure, boosts can often get you more than close enough, and that's how it was done on a lot of classics. But then, there's all the stuff that was recorded with a hot pickup into that boost and hot amp...