Are you looking for something to solely to satisfy Thai immigration or looking for something to both Thai immigration and cover you in case of ill health?
There are much cheaper option for the former. (Off hand my policy is < $400 per year. ) They're not terribly good for the latter but healthcare cost in Thailand is super cheap compared to the US anyway.
If you found a branch who is willing to talk to you, you're good.
From my experience I needed a certificate of residence, my visa document and my passport to open an account. Not sure if a Thai phone number was required but they took mine.
Note; every bank and every branch seem to do their own thing. A couple of banks turned me away cold and others were happy to help.
You need a residence certificate and a doctor's certificate beforehand. Both easy to obtain. Once you have them, make an appointment, go do some minor eye tests and they issue you with a Thai license. I forgot the cost but it's not much.
You're going to wait a little bit especially if there is a crowd. I think it took me about 2 hours from arriving to walking out with my driver licenses in Rayong.