Support got back to me again. They insisted this time however that the problem was that I had the game installed on a drive other than C:.

They completely skipped over the whole "game works just fine as long as I have manually terminated every running instance of Chrome on my PC".

Instead, they suggested that the game is fundamentally incompatible with running on any drive designation other than C:, and I quote:

Unfortunately, FINAL FANTASY XIV is not optimized to play on a different hard drive than "C". We understand this can really be frustrating, but even if you have played more than 2,500 hours on the selected hard drive, eventually there were going to be issues since the hard drive "D" is not compatible. Although the option to play using the hard drive "D" is not available, we will take this information and forward it as feedback to the appropriate department. While we cannot guarantee implementation, we would like to assure you that it will be reviewed by the appropriate party.
I refuse to believe that this is the case that the game is completely unable to run on a drive other than C:. This isn't the early aughts. No one makes programs deliberately to only run when installed on C: anymore - or accidentally overlooks basic data conservation practices of putting boot on one drive and data on another and makes it oops-not-compatible with innumerable consumer's setups. Not in a high production value consumer product they don't. So obviously this is complete nonsense. Absolute gibberish. Baloney. Hogwash. Gobbledygook.

Still would like either a refund or at least prorated game time for the last week and a half they've been running me around the ringer like this.