I learned to flyfish with my dads fly rod when I was a kid. His old reel wasn't really set up for reeling in one directio or the other. They didn't have drags on fly reels in those days (makes me sound older than I am), but in any case, you just turned the reel around and respooled. The reel had a clicker that worked in either direction, and your palm was the drag.
I think there are a lot of flyfishers who started as spinfishers, and that may be one reason for the switch to left-handed reels. Also, formerly it didn't matter which side the reel was on, because it was so rarely used. Only in the last few years have people's minds begun to change about whether you should play fish on the reel. Now, it's encouraged much more, especially for big fish.
I learned to strip the fish in, though, so in the heat of battle, I never remember I can play the fish on the reel, and the few times I have had someone urge me to, I lost the fish, because it surged and fly reels don't have favorable gear ratios that allow you to bring line in fast.