I've always had a thing for them old Enfields (got 3 in the house now, 2 of which are mine) but I voted for the 6.5mm just because its a flatter shooting round and better for the prairies IMO, the .303s just don't have the flat distance over 200 yrds I find. I shoot in SW BC which is fairly heavily wooded so a shot beyond 100 yrds doesn't generally happen, however both hit hard enough to take large game down under 300 yrds.
As for the stock and rifle, I don't think its ugly at all. I'd personally keep it on the mantle as a tribute to the old timer who carved it but I wouldn't be ashamed at all to shoot it. It has character and its unique, keep it that way.
I'd personally take the Weatherby anyways as its really not that large of a rifle, and once you're at it out there you won't notice the extra bit, and it's also probably lighter weight-wise. Plus, you'll get more power and velocity from the longer barrel which you'll need for moose and elk. Until I bought my Rem 700 .30-06 this fall I have hunted for years with a fully original 1942 .303 Mk 4 with an old POS 7x scope (which I think is damn near as old as the rifle!) on it and it drops deer just fine. I just wanted to retire the ol' girl before it got beat up considering it was bought by my gramps from the Vancouver Army & Navy Store and only had 15 rounds put through it since he bought it in '58, and I've only put 2 or 3 boxes through it myself. LR