While I think rhythm is a valuable skill, I don't think it's essential to fighting.
I'm a musician, so rhythm comes naturally to me, but in fighting I really don't follow any rhythm or pattern. When I'm grappling I like to always be working, whether for a dominant position or a submission, but I try to change direction constantly to keep my opponent on edge. If I ever find myself developing a "pattern" (I used to have a problem where as soon as I got the mount I'd go for a kimura) I try to break out of it. This normally surprises my opponent and gives me edge. So sure, you could call what I'm doing "rhythm". I call it tactical speed and force adjustment, relaxing then exploding.
In sparrng I do often work for a rhythm, but only to lull or "hypnotize" my opponent. One thing I find especially effective against boxers is getting into a pattern where I always step in and throw a double jab, then a right hook if it's safe. After doing that 4-5 times I step in with a single backfist instead, then give a straight right to the midsection. Against my muay thai buddy I tend to hang back and wait till he's moving towards me, then explode forwards and push him back with a flurry of strikes. It tends to overwhelm him, and when he tries to change direction he gets caught, even though he's learned to anticipate the move.
So it really depends on the type of fighter. Against some punk on the street I'd probably use a lulling rhythm and non-telegraphic strikes. Against a trained mixed martial artist I'd be more comfortable with constantly changing my pace.