I personally would consider a 338 Lapua or 338 Edge. These are modern cartridges with much more support and components compared to the .338-378 Weatherby. The Weatherby has sort of fallen to the way side and lacks support and components and data can be harder to find.
I disagree with you jud96. The .338-378 is very much alive and doing well. The same components available for any of the cartridges you just mentioned are out their for the big weatherby. Two brass source's with weatherby and norma and for data most reloading books have that covered. All one has to do is go to hodgdons web sight and they list several powders for the .338-378. They even list a relatively new imr powder,imr 8133, the big weatherby is a screamer and a power house. Does the .333-378 lack fan fare on this forum? Absolutely it does,but that does not make it by any means obsolete. Show me where I can find readily available data for the 338 edge,other than on this forum and maybe one or two other sights. Do I have to fire form the .338-378 weatherby? no I don't unless I have an improved chamber, which can easily be done. There are others on this sight that own the .338-378 weatherby and I know that if blue wants to go ahead with his new build on the .338-378 weatherby he will get plenty of support. I say go for it blue. I'm sure others will come forward. ctw
There are a multitude of details to discuss before building something on this case:
Find the action which will suit your use and measure the magazine area. This cartridge case is long right from the get-go so using higher BC bullets or solids will make this into a single shot real quick. There's nothing wrong with a single shot because you have the horsepower behind some excellent bullets.
Once you find the action, be sure to get a source for a high quality stock. Remember this is a bigger action which needs more space for the bigger inlet. Reinforcement is a good idea because of the recoil when the case is paired with a heavy for caliber bullet. Carbon fiber can be helpful in many ways.
High quality glass is always a good idea but be sure to get a high quality scope mount or rail and rings to keep it on the rifle.
Everyone has an opinion of barrel makers so pick one of the top 3 or 4 and you should be good. Be sure to consider your case and bullet combination. Discuss this with your gunsmith to determine the freebore required to accommodate the length and it might call for a custom reamer. Make dummy cartridges to send to the reamer maker.
The discussion of brakes is usually contentious with the division pretty close to 1/2 and 1/2, for and against. I use brakes on most of the heavy hitters for obvious reasons. Some though will perform better using a suppressor. Barring the cost and wait time for a can, there are excellent brakes on the market these days but muffs and plugs are a great idea!
There is a ton more but book length post are not useful.
One thing to consider is another cartridge, the 33 XC by David Tubb. No belt if that matters and it has just about the same capacity as the Weatherby. Brass and dies on Tubb website.