I ride a fixed gear bike with SPD clips and mountain bike shoes. Why they call this clipless makes no sense at all. They are clips!!
Anyway, it's no problem for me because 1) I have brakes on the bike 2) I feel confident riding with SPD pedals.
I would never ride a fixed gear bike with clips and cleats on the street. My opinion alone but that's suicide if you ask me. I know, I rode with those "back in the day" with my road bike before "clipless" systems came on the market. More than once I fell over at a street light.
Clips and cleats on the track? Yes, sure , it's a controlled environment with no cars.
There's nothing to be afraid of with modern "clipless" systems. Just give yourself some time to get used to your foot being "trapped" in clips with a wheel that doesn't stop before you launch yourself onto the streets.
And yes, every now and then I forget I'm riding my fixed gear bike because I do alternate with my multi-geared road bike. You only forget for about a millisecond though! ROFL.....