About five years ago a guy brought a .30-378 out to a couple of RNBRA matches. The muzzle brake on the rifle was technically illegal (because they are hard on adjacent competitors), but we let him shoot it and tried to place him off to the side.
He was quite a good shooter, and he ended up getting much better results with the rifle than I would have expected. I had formed an opinion of Weatherbys as being "flashy trash", but this rifle actually shot quite well. We weren't shooting long range that day, but at 300m-500m-600m it seemed to me that the rifle really was capable of shooting better than one MOA.
It was an impressive show, but not a very practical rifle to shoot. He later built himself a topnotch .308 F-Class rifle and fired it a couple of years, doing quite well in competitions.
.30-378 Weatherby is not used in F-Class (long range precision target rifle shooting). While it would be legal to use it in the "F/Open" category it would not be competitive. Believe it or not, it does not have as much performance as what is used to compete and win (typically a 6.5-,284 firing a 140-class match bullet, though in the past couple of years people seem to be moving to 7mm cartridges firing 180-class match bullets).
Part of the reason for this is that there are no good heavy .30 cal match bullets on the market, so any .30 cal target rifle will be handicapped by not having bullets as good as the 6.5mm or 7mm cartridges. Sooner or later this will change, eventually the good bullet manufacturers might come out with a 230+ grain long range target bullet (BTW Sierra has a 240 grain Matchking, but it does not count as a high performance bullet).
A .30-378 is in kind of an awkward spot, it's either too big or too small depending on how you look at it. If you are willing to accept a lot of cost and recoil in order to get high performance at long range, you're probably better off getting a .338 Lapua or similar. If that is not the route you want to go, then perhaps a conventional F/Open rifle (e.g. 6.5-.284 or .284 Shehane) or a conventional F/TR rifle (.308 Win or .223 Rem) would be the best bet.