While I would love for a server solution, I'm imagining that it is a logistics problem having to do with server bandwidth and storage space.
Just think of it.
\My Documents\My Games\FINAL FANTASY XIV\ (where I've read the stuff is stored)
Is 20MB and I only have two characters (the one I haven't touched in forever).
Now, if we assume that everybody used the server to store it, let's say 15MB for a single character (that other character is nowhere near high of level or has as many hotbars etc to save), and if we assume there are how many people on a server again? According to a March 2017 census, the American server with the highest character population was Famfrit at 290k.
So, you would have to realistically provide for ~300k to allow some buffer room. If each character is 10-15MB of data (possibly more, so let's say 20MB to be generous because when you program, you do generous numbers to make sure you don't overrun), then you need to provide nearly 6,000 Gigabytes or 6 Terabytes.
I dunno about you, but that sounds like a LOT of storage space to devote for UI settings.
So what do you do? Limit how long these last? Limit how many character(s) a player can transfer at once? Limit how many people can do this simultaneously? Then you start getting angry people who are angry that the service isn't good enough.
What I WOULD like to see, however, is for one to set an option so that the game uses a removable drive to store this data on, so that you could simply specify the path (like G:\ or something) which points to a USB thumbdrive. Boot up PC#1, stick the thumbdrive in, have FFXIV transfer data to this thumbdrive and tell the game to use this for the settings.
When finished on PC #1, you would then go to PC#2, stick the thumbdrive in, boot the game up, tell it to use the thumbdrive for this data and from then on, you'd simply plug the thumbdrive in anytime you played.
OR, you could also use a LAN server -- tell the game to store/use data on a computer running a server (you could use any cheap laptop running linux for this purpose, as long as it can act as a file server), that way no matter which PC you logged on, you'd always have the same data available without manually telling the launcher to re-load (or re-upload if you changed anything) from a specific source.