Realistically, while they could do it for orchestrion rolls, minions, or mounts—all things which are tied to account entitlements/achievements and so which can show the "Unobtained" or "Obtained" in the tooltip in game—they couldn't do it for outfits (which are just items).

I'm betting the store servers run separately from the account entitlement servers; if they just don't have a good way to query the data in an efficient manner, it would explain just not using it. It would also explain why it can take so dang long for things to show up in game at times. But yes, I agree, it would be really nice if it showed you "you already have this" for the characters where that information hypothetically exists.

Thankfully there are sites like FFXIVCollect which provide a checklist of all mounts/minions/orchestrion rolls/etc. which you can check things off on, but it would be nice to not require a third party site as a sanity check...