Plenty of information so far. I'll add a few thoughts. I would try to shoot someone's 375 if possible. Do you know your recoil tolerance? What other larger caliber rifles have you fired? If you have some details of past rifles you have fired you can put some data into a recoil calculator to get an idea of what you might need in the way of weight for the rifle and if the 375 is a good idea for you.
Calculator:
Recoil Calculator
I have a 8 lb 375 Taylor which is a 338 win mag necked up to 375. It is capable of getting a 260 accubond up to 2775 fps. That will wake you up from the bench. I chose to load the same bullet at 2650 fps with much more tolerable (for me) recoil.
If you are going to build from scratch you might want to consider making a 375 Ruger. The shorter case will fit a Rem 700 magazine better. Another reason is this shorter case will lend itself to reducing the load better than the longer 375 case.
I don't think a muzzle brake is a good idea for the 375 H and H. It won't work as well due to the expansion ratio with such a large bore size, but it WILL add more noise. You should be able to find a load that appeals to you without using a brake anyway. You can work up the loads to hotter/faster performace over time.
While I can understand the appeal of the classic 375 H and H, I cannot see building one when there are some great options now available. We all have our opinions. Your choice.