good question and having ridden fixed a bit I still don’t know the answer.
i picked up a cheap fixed wheel to stick on an old racer to dip my toe in the water and was deeply underwhelmed. Its like singlespeed really. Like someone further up said, no zen experience, just single speed but perhaps slightly easier to get over the deadspot at the top of the cranks when going up a hill. Oh and possibly the odd lovely brief moment when you see something up ahead, say red light, and instead of reaching for the brake you just pedal slower and drift to a halt.
Track racing looks fun, but i think despite rather than because its fixed.