Efficiency is less the issue for me on fixed gear. Sure it is nice to gain whatever but really for me it is about foot retention. I have ridden both on flat pedals and clipless pedals on my fixed gears and I feel vastly safer clipless. I have gone down some hills where my feet slipped off and it was tough and slightly dangerous trying to get them back on. However you could potentially achieve some foot retention with some pedal straps and maybe a nice grippy pedal (like an MTB pedal with adjustable studs) but I don't like single sided stuff because a lot of times it trends toward the side you need pointing down.

I always use a brake and have no reason not to do so! The only time I might say forgo a brake is if I was doing UK hill climbing races but even then I would probably have brakes for training and would just use really light brakes and levers and housing or of course at a velodrome where brakes are not permitted but I don't have a velodrome around me and while I would love to ride around in one I probably wouldn't make a schlep for it.