Turns out the errors I encountered were entirely related to the "new" SQL Server driver that ProGet 2024 was using. Although this was in beta for well over a year, it was "only" in production for a few months by the time we incorporated it into ProGet. This driver issue impacts anyone who uses .NET on Linux.
As I mention, I think it's pretty clear that Microsoft has effectively abandoned SQL Server. It's one thing to release something this low-quality and untested... and another to have it linger with issues like this in production for months. We will be moving to Postgres, which do not have these endemic problems.
In the meantime, I have instituted the "one year rule" for Microsoft products - meaning, unless it's been shipped in their general release channel for a year, we will not even consider using it.