Approaching the kernel year-2038 end game

You may be lucky that they don't use time_t in ways that cause an undefined integer overflow in 2038. However, since you don't have the source, there is no way of finding out whether the programs are safe or not, so the assumption would be that they are not.

The plans for glibc are to keep compatibility support for the old symbols, so the application will still work correctly before any overflows happen, so it's likely that you are safe until some time in the 2030s when it has set the first timers long enough into the future that they overflow (e.g. key expiration dates are often months or years in the future). Again, to be sure you'd have to inspect the source code or ask someone who can do it, but that would not be that different from asking them for a rebuild against a C library from the 2020s.