The Concours/FJR seem like what you're after. If you go FJR, avoid the auto-clutch version that was offered for a few years. They're relatively uncommon anyway, but it was a complex system that wasn't known to be particularly reliable.
The Concours14 has a little more aftermarket support, more power, and a 6 speed. It's getting a little long in the tooth and is pretty spartan for features. It's been essentially unchanged since it came out in 2008 aside from one very minor update in 2011-12 or so (heated grips were added, and they moved the cubby from on top of the tank to the side fairing- that's about it). ABS was optional on older models, and it became standard at some point, bringing KTRC (traction control) with it.
The FJR recently got a 6spd in the latest update, as well as an optional ES (Electronic Suspension) option. The FJR also got cruise control sometime in the last couple years. Some people couldn't care less while others feel that's a huge bonus over the C14.
Both have electric adjusting windscreens and 100K+ mile engines. To me, they feel pretty much identical aside from the Connie having a bit more oomph to the engine, although the FJR is by no means a slouch and will gladly whisk you deep into go-to-jail speeds if asked.
GSes and other ADV style tourers to me split the difference between something like a Concours and a Goldwing. ADV bikes personally I find more comfortable than sport-tourers with a more neutral/upright seating position. Connie/FJR still put you in a bit of "attack" forward lean, even with risers. It's vastly better than a supersport, but for a 500 mile day, you might start to find it tiring.
Thing is, the BMW1200 boxer I just find to be hopelessly boring. It has the personality of a kitchen appliance, and power can best be described as "adequate". Not thrilling, not dynamic, just "adequate". It's comfortable, smooth, and a perfectly capable touring mount, but eh, it's just dull as dishwater to me. BMW cars have the reputation of being "businesslike" and "stuffy", and their bikes echo that at least to me. The Multistrada 1200's have a far, far more entertaining engine, and reliability is actually pretty solid. I've seen early 2010's models going for within your budget, but plan for a $1500 service every 15k miles.
I'd suggest a FJ-09. It has the upright and neutral seating position of an ADV, the jewel of an engine from the FZ-09, and they can be had secondhand well within your budget, probably with a new nice extras. The "cons" are while it has a fantastic spread of smooth torque, outright speed isn't particularly fast- not gonna flatten your eyeballs when you wind it out, it has an IMO annoying/unnecessary governor at like 110mph, and stock suspension definitely is more conducive to "touring" than "sport". It's an ADV bike, without the off-roady pretense.
If you don't mind looking older and cheaper, a Honda ST1300 can be had for next to nothing, especially if you find one with a few miles on it. They have good power (not C14 fast, but not too far off), and the great V4 sound. Still very capable, bulletproof bikes that can be had for bargain prices.
If you consider yourself more of a sporty/aggressive rider who really wants to rail corners and crank the speeds up, I'd look toward the C14 or FJR. If you place a little more focus on comfort and less on speed, look toward the FJ09.