Nothing wrong with a pump for hunting... I prefer a bolt gun for accuracy at the range but a pump will do the job on a deer or elk just the same plus has the quicker action but costs less than an auto.
I've got my Dad's old 760 pump '06. Took it deer hunting for the first time with it. While I will still hunt with it in the coming years I learned a valuable lesson with it. I found a V-rest in a tree where I laid the gun in for a steady rest. I took my first shots at the deer and had to get out of the rest every time to load another round; yes it took me three shots. Anyhow, I discovered that I would rather have a bolt gun while in a rest or on a bipod.
I had a remington 760 pump in .308 years ago.
It was a good rifle. But If you plan to reload your own ammo a bolt is preferable.
Also in a pump you might want to avoid the light magnum type loads , not for safty reasons so much as the case might expand and cause your pump to stick. The remington pumps are often very acurate. But in general a bolt will be more so.
Had an early 60s Remington pump, but not for long. Recall zeroing it
in, was painfull due to the stock configuration. At that time two
were available, rifle and carbine. I had the shorter barreled one.
Accuracy was OK for hunting, but bolt is so much better that I
never went back. Nothing wrong with them, as they have been
made a long time. Just not user friendly for me, and can't see an
advantage unless you like them, and are a great buy, I perfer a bolt.