There's no magic to fix gears. No hidden benefits (at least from a physics standpoint). But they are 1) a blast for some of us, 2) a way of life for some of us, 3) very efficient for staying in riding shape on limited miles or time and 4) do a really good job of smoothing out pedaling, especially if you regularly go down hills that push your comfortable RPMs.
I ride fix gears for more than half my miles. I often change gears, both at home before I leave and on the road. I insist on having near velodrome/competition level reliability, so all my riding is on 1/8" gear. All my gear changing is done with stops; loosening the hub nuts and changing cogs (and on one bike, chainrings). Not very different from what was done 100 years ago. (At 64 yo, and 100k fix gear miles, neither I nor my knees have to prove I can still get up 10+% grades on the 42-17. And going down very long descents is much more fun in gears like 42-12 or 42-13.)
Part of the challenge of riding real hills on fix gears can be the decision to stop and change gears or not. A cost/benefit analysis. Stopping is lost time that may well pay off in spades hours later on a long ride. (It also allows me to do rides I simply could not do without the gear options. The hillier Cycle Oregon rides for example.)
Now, if more of us rode fix gear and more of us changed gears, maybe, just maybe, we could become mainstream enough that a creative mind would figure some of these options out (fix-free while riding, really good fixed hubs, etc.) We'd all win.
Ben