I don’t know when or why this feature was added to the game, nor do I know how broadly it was applied. But I noted today while fighting Amalia in Dalaran repeatedly to level my pets that she is healing my pets after every battle and has dialogue to go along with it.
Blizzard, if you’re reading - thank you. Whoever thought of this, I want to buy you a cake.
10/10 A+ 
She doesn’t heal your pets. That’s always been her line when she is defeated.
Were you using the Ranishu Runt and Voidskimmer strat from Xu-Fu? There is some built-in healing in the strat and I only needed a few bandages for the 3 or 4 hours I spent leveling pets. Ranishu runt heals itself some after attacks due to being humanoid, and the Voidskimmer cleansing rain heals its allies. I just use the built-in heal pet button occasionally if I have bad RNG in the fights over time and the Ranishu takes a lot of damage. This is my favorite tamer to use for leveling my pets. I don’t use scripts so it helps the strat is very simple, and you get XP for your character (unlike Squirt).
You ever need a stock of bandages, just watch the daily Pet WQs on SL, DF and Khaz Algar. Couple years back Blizzard must’ve decided players got too many useful drops from those WQs, started padding them with bandages. I realize bandages are useful, but getting 17 to 20 per drop, especially with alts working them too, they add up fast and are WarBank shareable. At least the stacking on them got upped to 5,000 per stack.
Earlier this week was an almost insulting day, the 4 mainland DF pet WQs and both DF patch WQs all dropped bandages.