Between the Bergara and the Browning, I would definitely go with the Bergara. The X-Bolt is an okay hunting rifle action, but it just feels off to me when I shoot one. Also, the stock trigger is not good - definitely needs to be upgraded, but options are limited. Honestly, though, I'm also not a fan of the base Bergara, and there have been a fair number of reports of issues with Bergaras in general in long-action magnums. They work well with smaller calibers, but it seems like they didn't beef up anything for magnums and they just have issues with the added stress.
I am a fan of the Bergara Premiere actions, however. I used to own one in 6CM and it did exceptionally well. They cost a fair amount more than the base Bergaras, but they are essentially a whole different rifle - upgraded stainless bolt, upgraded trigger, upgraded action. As
@Baron23 mentions, a Tikka is also a great alternative.
Also, going with a magnum like a 300 PRC out of the gate is a) somewhat overkill for most of the shooting you'll be doing, and b) is not a caliber I'd recommend as a first long-range rifle - it takes very solid fundamentals to get it to behave.
If you're mostly going to be shooting at 1000 yards, pretty much any high-power short-action round will work. For work at a mile, depending on the conditions around the target, wind, etc., something like a 6.5 CM _can_ work, but it's not ideal. But, likewise, using a 300PRC at 1000 yards is not ideal either - more power than needed for the range, more expensive per round, tougher to control.
As for scopes, a PST Gen 2 will more than get you to a mile because you can use a mount with, as an example, 20 MOA built in, and/or a rail with, also for example, another 20 MOA. However, I wouldn't recommend it for that distance. The glass isn't great. When I had one, I tried to use it at longer distances and I had my share of eye fatigue issues. At 1000? It works just fine - and is a pretty good bargain for what you get. Scopes that will treat you well at a mile are a class up from those that will do okay for you at 1000 yards.
Recommendation: Get a rifle chambered one of the more potent short-action rounds, get a scope in the class of a PST Gen 2, get good at shooting at 1000 yards, then play at a mile and decide what to do from there.