Tip: to track the play time of the game with any version, type "/playtime" on chat window.

The fraud they're trying to avoid is this: people with bot accounts would buy the game on Steam to run with the bots. They would run the game outside of Steam while leveling the bots, making gil, getting banned, etc. Then they would ask for a refund on Steam by saying they didn't play the game. Since Steam wasn't used to run the game, it won't have tracked the time they spent and the refund would go through.

To avoid this refund scam, they're gonna force Steam versions to run through Steam. Makes sense, right?

There seems to be a workaround though. You can run a Steam account either from non-Steam installation or from Steam folder but without Steam by running a shortcut to "ffxivboot.exe -issteam". Not sure for how long it's gonna work, as it seems to be a temporary solution while they settle everyone to their respective installers.

I imagine that they might have a similar shortcut allowing non-Steam accounts to run on Steam. Contact the customer support and see what they can do for you.