Ok, more data
I put my Tmobile SIM in an old Samsung Galaxy S4 that I had on hand - the one I had used to see if it was really pin-locked - and I think I found the root of the problem.
I had activated the SIM and tied it to my account over the phone. Despite that, Tmobile wants you to go through an on-device activation process to set up some sort of management account - not all that different than what the BRCK does, to be honest. I suspect this process is blocking the BRCK from using the data stream.
I signed up, and landed on a management page - but there I am temporarily stuck. I can't get to any internet site - every web page results in a redirect to the Tmobile activation screen, even though I've successfully completed it. I'm going to give it a few hours in case it takes some time before I give them a call. But clearly, this one isn't a BRCK problem. My guess is that I'm provisioned like a tablet, and in Tmobile's world, there should be a user at the end of that.
I think the BRCK folk really need to set up either an area of the Forums, or a Wiki, or some such, for dealing with carrier-specific activation and setup issues. It's likely that this is where a LOT of first time stumbling blocks are going to lie.