Your description of the way of fixing a flat on a fixed gear sound much more work than the deraileur way. You disengage the handbrake if you have rim-brakes, open the quick-release for the wheel, bend the deraileur outwards a bit and just lift the wheel out. The gear is already at a low gear (does not have to be smallest) since you just got a flat.
Reinstall by moving the casette back inbetween the chain on the same cog and just drop it down and flip the quickrelease back to locked. Connect the brake again if you have rimbrakes. No need to fix chain tension.
I guess it all depends on what you are used to do. I had soooo many flats last year when the spikes started eating through my one year old schwalbe wintertires. 220 spikes makes 220 holes. Had to give up on fixing the holes and change innertube until I could get new tires. Not schwalbe this time.
Don't think tubeless would do any difference here though, see enough people get flats with those too and as someone else stated, the glue can get pretty messy.