SO with this idea, I can craft tier 3 level 20 in any settlement (once they are trained and I have mats), even Thornkeep, right?
No, only in settlements that have the crafting facility you need. (Thornkeep does have all the facilities)

By the time we have L20 crafters, the facility ratings of various buildings will be varied enough that you will probably want to find a settlement that has a high level facility, since facility quality has a significant effect on crafting time.