I haven't had to deal with this personally, but I'd imagine you could keep two copies of each MXD,
one with the generic browse user password, and the second with password retention disabled.
Users grab from the second pool, but that pool is created via scripting from the first pool (which
is closely held). The publishing proceess would need to be amended, but dealing with a thousand
of anything generally requires a "policies and procedures" doc.
You might also publish only with a shared read-only user, so that users would be encouraged to
use their own logins (or time-limit the shared login to force use of their own logins in frustration).
There's probably dozens of solutions, though not all of them are practical at all sites.
If I had them to spend, my UC "Esri dollars" would go to more robust transition tools.
- V