I did call Eservices and this is what they told me.
The Eservices technician asked about how long it's been since I accessed the Transcript Delivery System. I said, probably not for a couple of months (although that's unusual for me -- I usually access it about once every couple of weeks).
The technician then checked my account and she mentioned that my EFIN number had expired or was no longer valid.
I said, yes, that number expired years ago -- apparently because I have never used it. (As I recall, issuance of an EFIN was just part of the process of setting up an online account for a practitioner about eight years ago -- regardless of whether the practitioner would be using the EFIN to file tax returns.) I explained to her that having a current EFIN has never been a requirement to access the Transcript Delivery System.
She said, yes, but the IRS did a "clean up" of the Eservices system in March. She said that now, a practitioner needs a current EFIN to access the Transcript Delivery System -- again, even though (A) accessing the Transcript Delivery System merely to obtain records and (B) using an EFIN to file a return are not at all the same thing.
So, she had me re-apply on line, with her on the phone. She said that once I receive notice that the re-application has been approved, I will need to call Eservices back and tell them to go into my account and input a special code that will specify that the account must stay open, regardless of whether I am using the EFIN. (Presumably, Eservices will also re-instate the EFIN that I had years ago, or will set up a new one for me.)
So, hopefully that solves the mystery.
Perhaps it didn't occur to the programmers who "cleaned up" the system in March that having a current EFIN shouldn't even need to be a requirement for merely accessing the Transcript Delivery System. Also, it apparently didn't occur to them that there might be practitioners whose EFINs had expired years ago and who would now suddenly not be able to access the Transcript Delivery System once the system was "cleaned up". They should have advised practitioners of the effect of the "clean up".
Edited for a typo. LW25