I was just reading a Quora post on the longest overland trip in the world and it mentions crossing from Georgia to Turkey at Kartsakhi. I've crossed between these countries probably half a dozen times and used both crossings I knew of. But the last time was in 2012 I think.
So I did some searching and found an article on the opening of the border checkpoint, but it doesn't mention whether people who are neither Georgian nor Turkish are permitted to cross.
(It's only a couple of months since I turned up at a border crossing from Cambodia to Vietnam that everybody thought I could cross and found out it's only for locals. So I'm pretty sure this is common for crossings in various parts of the world.)