I was curious so I had to go and test this, and it turns out it doesn't. Not really anyway.

I checked some sites and it seems like Flaming attacks halves how much healing its target takes but it doesnt eliminate it. Flaming Attacks does counter the hell out of "Regeneration effects" but that's always been the case.

I ran two sims on Hippogryph Knights vs Bloodthirsters in equal numbers. The ones running the healing performed better.

Although I did also discover that I might have been off by quite a bit when I said this would be an "easy" win. Bloodthirsters and their bonus to Large make them eat Hippogryph Knights a lot harder than I was expecting.