To answer your question, yes the 375 works well on just about everything that walks the earth. It is one of the most veritile cartridges available. As long as you use the correct bullets.
That said, the first season that I was using mine I was using the cheap federal classics 270 gr soft point. I though that since they were 270 gr that it would be a tougher bullet than the 180 gr in 30-06. Wrong. These bullets acted like ballistic tips and exploded on impact. Huge damage and it didn't matter at what distance either. I shot 4 deer from 80 yards out to 275 and the effect was the same for every bullet. Lots of damage.
The next season I changed and went back to one of my favorite bullets, trophy bonded bear claws. I tried the 250 gr this time and could not have been happier with the results. Just like I expected, nice controlled expansion. That season we shot 6 deer and a badger with the 250 gr bullets and every one was perfect. Nothing excesive in the way of damage. Even the badger, which was shot at about 60 yards only had a quarter sized exit wound. You can't really see it in the picture but there was little damage done to the hide.
I would not hesitate to use a 375 on deer or anything else for that matter. It is the best way to get use to a rifle is to use it all the time. That is why we have been hunting with ours for the last couple of years. We are preparing for our trip to Africa and what to make sure we can handle our rifles.
If you have any other questions don't hesitate to ask.