Ok, I think we were able to solve the mystery 
As @Hifihedgehog pointed out, new Twitter developer accounts start out on the “Essential” plan and are limited to the v2 API endpoints. Our built-in Twitter login currently needs to access v1.1 endpoints. Thankfully that’s not a problem!
There are two important things to note on this chart at Twitter API | Products | Twitter Developer Platform
You can gain access to the v1.1 API for free under the Elevated plan. You simply need to apply from your developer dashboard. I found it very quick and easy. Discourse Twitter logins worked flawlessly after that 
I’ll update the OP to include this newfound requirement.