Are you SURE you like detachable mags? Why have something else to potentially lose in the field, that serves no purpose whatsoever? That's my beef with the X-Bolt (and many many others today).
So, yes, Winchester 70 or CZ 550. Or, find a T/C Icon Classic or a used A-Bolt II or a used Sako 85 or used Weatherby Mark V - at least, those would be the suggestions in the general budget range you suggested. But there are many other good choices in higher and lower general budget ranges.. you willing to buy used, or new only?
As for chambering, 7 rem mag is not a good choice, in my opinion. Very expensive, overbore, inefficient; therefore unnecessary recoil. Get a .280 Rem or .280 AI instead. Even a 7mm WSM or 7mm RSAUM would be better than 7 RM, due to lack of a belt. 7mm RM is not terrible choice, mind you, but certainly wouldn't be in my top 10 choices of chamberings for an all-purpose large game hunter. I'd even pick 7mm-08 before 7mm RM.
And I will give contrary advice to above - don't get a 26" bbl in a hunting rifle ... get a 22" or maaaybe 24"...long bbls are surefire recipes to plug the bore with mud crossing creeks, get hung up in more brush, etc.
If this helps you.... I don't have a Browning, and I've never had a Sako, Mauser M12 or other really high end, but I have had Savages and several Rem 700s (and still have one) -- and I can tell you, that among turnbolts, the pride of ownership stemming from fit, finish, function (performance), and features is the greatest with these of mine,
in order:
-T/C Icon Classic (not the Icon "Original Medium Action")
-Winchester 70 (recent FNH /South Carolina manufacture)
-(tie) Weatherby Mark V (Ultra lightweight)
-(tie) CZ 550
-Ruger 77
-Rem 700 (customized)
So got that? Even a highly customized Rem 700 is less pleasing overall than a Winchester 70 etc., straight out of the box. Nothing wrong with the Rem 700 - it just lacks a little bit of the refinement (smoothness of action, great trigger. 3-position safety, "controlled-round-push-feed", etc.). The new Win 70s are the real deal in "mid-high-end" value, IMO, now that the Icon Classic is no more (and the A-Bolt II is no more).
Another thing to consider, if you haven't already, is that the Medallions and White Gold Medallions, though very beautiful - you have to ask yourself - "Will I be cringing every time the stock scrapes a tree or briar?" or will you be ok with that? I would be cringing, so I like more "workaday" plain wood or fiberglass stocks, that you almost WANT to put scratches on, to give them character.