Re: .30-378 Wtby Mag vs .30-338 Lapua Mag vs .300 RUM vs .300 Jarrett
I honestly don't think that there has been a cartridge suggested which wouldn't get the jobs you've outlined done (that is, all should have both the pop and the accuracy you desire, given a proper build).
If I were doing the build, I would weigh heavily the hardware and logistics practically required to make each chambering tick. The 338 Allen mag is quite a cartridge, but it will require a specialized, oversized receiver to make possible. They are cool no doubt, but I see their niche as being more of an extreme circumstance long range hammer, something to get after you have the game figured out, to extend your range on targets and varmints (buffalobob's 7AM is ostensibly a 1500 yard elk rifle, you planning to shoot elk or deer further?).
For me (not everyone feels this way, but I think Kirby's own Idaho hunt story is as good an argument for it as any), a dedicated big game rifle should most likely be a repeater. Only some of the bullet/chambering combo's suggested are easy to make into repeaters on standard actions. A 338 Edge, in repeater guise, is an easy rebarrel on any plain old Rem 700 Ultra mag action. The 30-378, for example, would most likely require either a MK V, or some form of oversized custom, to qualify as a repeater.
Bullet availability is legitimately a factor. The bullets so many are having great luck with in Kirby's rounds (Wildcats), are only available from Richard (unless Kirby happens to have your flavor stockpiled). While they are obtainable, to suggest that they are as easy to procure as a .308 or .338 Accubond or Matchking wouldn't be true. I wouldn't let this stop me from getting a 7AM, but I would want to be aware of it in advance, and take it into consideration.
Finally (chambering aside), for an all around big game long range gun, I would not want to go too heavy, overall. Many would steer you heavier, and they wouldn't be 'wrong', but I wouldn't want anything much more than 12 lbs, and would consider under 10. Many be many times well after you build it you may want to take it a mile from the truck or more, and a 20 or 25 lb rifle could make that trek a royal pain, and ultimately discourage you from taking them. While true ‘heavy’ guns have their place, a reasonable weight gun will ultimately see far more action. You can build a 'carryable' weight gun which still offers all the horsepower and accuracy you need. Check Kirby's green lightweight 7AM for an example. Lot of info here, lot of directions, options. Hope we have helped you identify some them and sort them out. Good luck.
Empirical results rule!