I'd go with a 7 mm. I hunted with a 7 RM for years and loved it - since went to a lightweight 6.5 PRC and a 9.5 lb 300 PRC for my primary hunting rifles. Deer or smaller 6.5 PRC, elk or bigger 300 PRC. Both have enough for all non dangerous NA game in a pinch and not excessive for smaller game.
For versatility, panache, ammo capacity, modest recoil I'd go with a 280 AI - even though I don't own one. A 7 RM or 7 PRC is nearly as good at the cost of slightly more recoil.
In fact, I recently purchased a Blaser R8 in 375 H&H for my DG and 'traveling rifle'. Am leaning towards a barrel in 7 RM for my hunt the world combination. Would be a 280AI for sure, but the 375 and 7 mm RM use the same bolt face and magazine which simplifies things for sure.
The 6.5's have their adherents, but they are a little smaller than ideal for game such as elk and moose. IMO, the most versatile calibers are 7 mm and 30 cal.
When I got my 7 mm 10 or 12 years ago, I read that if 90% of your game is smaller than elk choose a 7 RM, if 90 % are elk or larger choose a 300 WM. I think it's still good advice, and anything similar in performance to these two cartridges are great, versatile options.
But go buy a 280AI. Have a look at the Weatherby Backcountry 2.0 and the Browning Xbolt 2 if you're looking for rifle selections