Go with what you have - and know and like.
The airhead is probably marginally
better off-road - but that's arguable. It's certainly lighter and simpler, but they are also getting OLD. A major rebuild is a good idea, replacing all the old rubber bits etc.
They have their flaws, all of which are well-known. That's a plus - easy to sort and prepare for and fix.
But the R11xxGS is also pretty well-known and reliable, so personally I wouldn't worry about it. DO your research
and make sure you have covered all their known flaws - e.g. needs Touratech hard parts, rear subframe cracking etc.
Then just go. And have fun!
One thing to remember - for a lot of people the difference in cost is VERY important. And it's not just the initial cost, but the on-going maintenance costs are higher on an oilhead - you can't even adjust your own carbs. And most importantly, the carnet is much more expensive - us North Americans have to put up a bank bond for as much as three times the current value of the bike for some countries. For some people that's their entire trip money if they had a new R11xxGS!
My personal recommendation for most people travelling solo - an F650 or KLR or similar - NOT a 1000cc plus bike, air or oil head - it's just NOT needed.
Two-up you're back in air/oilhead territory.
And of course when travelling with two bikes they should absolutely be identical.
As for your question "What can be done with an airhead that can't be done with an oilhead?" - that's a tricky one - STOCK STANDARD there is a huge difference in weight - and off-road in the horrible gnarly bits the airhead will win. BUT load them both up with touring gear, and neither of them can go anywhere near they did unloaded, so the differences become minor - if anything I think the oilhead is better - it seems to be much more forgiving and requires less effort to point it where you want to go. The extra weight is only a problem when you screw up beyond hope of recovery. Your ability will determine how often that happens ...
If your ability is even slightly in question - in other words you DIDN'T win the Dakar last year - take an off-road riding course, you'll be a better rider for it and glad you did.
So, fwiw - go ride it. If a guy can ride a Gold Wing around the world, and someone else can ride a Fireblade AND an R1 around, any GS will be brilliant!
Seek, and ye shall find.
One world, Two wheels.