it is all online now, but each Thai embassy in that country processes the applications for their region. They aren't all centrally processes at one location for the whole world. You might not be aware of that detail, as many aren't.
buying a visa is no guarantee you will be let in if you've been trying to live here long term as a tourist. Even if you bought a tourist visa they can still refuse you entry.
It is very simple at this point, but could change in the future. If you have filed income tax in the country you're suppose to, and can prove it, then Thailand doesn't really care at this point.
you see on that list of required documents, where it says identification cards, then after it house registration? That is how they identify a Thai citizen.
Just go to the local amphur with your wife, and ask them how you get a yellow book, and pink ID card. They are not your enemy. That's what the bank wants so do what you can to get it.