Personally, I am recoil sensitive. I don't generally feel recoil when shooting at game, but higher recoil guns are not something I am eager to practice with and it is easy to develop a flinch.
I've never thought I was all that recoil sensitive. However, that might be because my dad was probably the most recoil sensitive person I've ever known. Dad carried a semi-auto .308 Winchester (and used it very effectively) for big game hunting for most of his life because he believed 30-06s and larger cartridges "kicked too ______ hard!"
Dad also selected a semi-auto .308 Winchester (just like his) for me for my first big game rifle (when I was about 14) because he knew a .308 Winchester was plenty of gun for big game anywhere in the lower 48, and he didn't want me to develop a flinch. I'm sure it was a disappointment to Dad when later on (back in the early '80s) I "stepped up" to a 30-06. And Dad would absolutely be shaking his head now that I've become so fond of my 30-caliber magnum (my beloved .308 Norma) if he was still around.
On the other hand, my 5'2",120 lb. wife uses (also very effectively) a custom 7mm Rem Mag for big game hunting. However, as I've said before, we load it way down for my wife to practice and occationally shoot rockchucks with in the summer. Then when fall and deer season comes around, my wife uses a recoil shield to fire a couple of full-house rounds just to make sure her rifle is "on." And then like you said, she never "feels the recoil" anyway when she's shooting at big game.
Besides that, even the "full-house" 7mm Rem Mag loads my wife uses are more like .280 Remington, or 7mm-08 "+P" loads than 7mm Rem Mag loads. The fact is, the only reason my wife uses a 7mm Rem Mag for big game is because she wanted a custom big game rifle built around a Model 70, pre-64 type action, and the only used rifle like that we could find happened to be a 7mm Rem Mag. We knew we could "load down" a 7mm Rem Mag to 7mm-08 (like she had been using for big game since the 7mm-08 was first commercialized) levels anyway, so that used 7mm Rem Mag is what we bought.
If my wife goes before I do, I'm going to have one heck of a time selling that rifle. Sentimental reasons for one thing, but also because it's one weird looking rifle IMO. It has a long, 25" barrel, and an almost unbelievably short stock - to fit my 5'2" (with short arms to match) wife.