I currently own two Browning A-Bolts one in .308 and another in .223. They are excellent rifles and IMO they offer a lot of value over a Ruger or Remington bolt action rifles. As you know the Browning barrel is free floated, the trigger is adjustable, and the 60 degree bolt action is fast and smooth, also I like the position of the safety.
The new X-Bolt is nice. The Trigger is better, the X-Bolt Action allows you to remove a live round from the chamber while the rifle is still on SAFE, unlike the A-Bolt II, and finally the X-Bolt has a true detachable mag vs. the hinge floor plate mag like the A-Bolt.
So between the two I think the X-Bolt is certainly an improved version of the A-Bolt, but I am too happy with my A-bolts to make a change to the new rifle.
As for the BLR and the BAR. I used to own a BLR in .308 and it was a rifle that I really wanted to like, but it was major disappointment. Mine was the older style with a 20" barrel and a straight stock and had the barrel band. From what I've read that barrel band on the older BLRs tends to hurt the rifle's accuracy. My BLR shot about as well as a Ruger Mini 30 in 7.63x39
Not good.
The BAR on the other hand, more importantly the new Browning Shortrac or Longtrac in 30 06 is a fantastic shooter. The old BARs are heavy and can be finicky, but the new BARs are light weight and shoot MOA.
I set up a new Browning LongTrac in .270 with a Leupold 3.5x10 for a customer a few years ago and I was shocked at how well that rifle shot. I am a big fan of hunting with semi auto rifles, I like military style rifles like the HK-91 or AR-10 because they are reliable and accurate vs. a Remington or older BAR.
My Uncle bought a Benelli R1 in .308 and spent a lot of money on it and to be totally honest that new BAR Shortrac is a far better shooter than the Benelli R1 and costs about $400 less.
I say if you want a great deer rifle the X-Bolt is a great choice, if you want a the best lightweight semi auto deer hunting rifle with BAR LongTrac in 30 06 is your ONLY choice vs. the Remington 750
or the overpriced Benelli R1.
Hope that helps, stay safe and thank you for all of your work over there.