I swapped between disc and shadow in ICC on my priest. A lot of fights didn’t need a 3rd or 5th healer so I changed a lot. It really wasn’t a big deal at all. I don’t know why it would be a big deal now.
Before Dual spec, it wasn’t uncommon for us to bring an alt in and out of fights for gear.
You do mean if you’re running M+ right? If you’re doing heroic/normal dungeons, or M0, you can change your spec partway through the dungeon if you want. I’ve swapped from healer to tank in a normal after the tank dipped just to finish the dungeon a ton of times.