I took 2 6 pt. whitetails this year with my Zastava z98 in 375 h&h. Both with 260 grain accubonds. It worked well, both fell like they had been dropped out of an airplane.
One was shot just a little in front of the "ideal" shot placement in the boiler room, so it went through some of the front shoulder meat, but there was no significant meat damage (no more than a 308 with similar quality bullets). I recovered that bulled from an oak tree on the other side of the deer, it went in to the tree about 4-5 inches, sideways. It was perfectly mushroomed, and it had lost about 20% of its weight.
The other deer was running through the bush, and stopped dead when I expected him to continue, so he was shot through the neck. Clearly no significant meat damage, neck was broken. Bullet was not recovered.
What I did notice though is that although it is alot more gun than a .30-06, there was no significant on game difference when talking about whitetail deer. I've got a Browning FN Safari in .30-06 and for some reason it just knocks deer over when loaded with 180 gr. partitions. I've never had a deer run after being hit with that combo. So I guess what I am saying is that when something kills game, it kills game. It can't be any more dead than dead.
Is the .375 too much for most NA game? Well that depends on how you define it. Is it somehow less ethical to use more gun than is necessary? I doubt many will make that argument. Is it more than is necessary by a large measure? Probably, but that might make it inefficient, that doesn't meant it is "too much." Is it so big an scary you will separate your shoulder/detach your retinas/rattle your teeth out? That depends on the shooter, I don't find it to be unplesant to shoot, but after a box off the bench my shoulder might be a bit tender. Still, this doesn't make it too big for NA game, if it is so big it is injuring you, it is too big for you in general, not the game you are shooting.
Also, as is always said on these threads, make sure you use bullets of good construction (which I've found most .375 bullets to be anyways.. not too many varmint grade balistic tips made for the .375). My deer both had minimal damage, no blood shot etc, but another guy at our camp shot an 8 pt with a .300 sav, right in the wheelhouse, hit a rib, and the whole entry side was bloodshot and bruised up, and we lost a bit of meat off that deer. He was using some old dominion 160 grains. It was real surprising.