Honestly if I had to guess it was never the intention that people swap covenants to do this daily (because as you said it had a cooldown). It’s just a quirk of the relaxations given to the swapping system that you can now.
Otherwise Shadowlands was still an xpac of bad systems and grinds, for sure.
Oh and by the way, I can’t remember if you get presented different quest givers or possible rewards by swapping in and out of warmode…but for some reason that rings a bell. I’d test it!
Here’s an old guide that probably holds up today: A guide on Kothia daily quests transmogs. : r/wow