You won't find a file with the library's name because even if it's there, it will be part of the shared dyld cache mentioned in your original message. Similarly, you won't find any other library, including the standard C library, on your system, but that does not mean they can't be used (if the standard C library couldn't be used, not a single program could be linked or started). In other words, looking for system library files by name on a macOS 11.0 does not tell you anything about whether or not it can be used.
To debug the issue, try starting by printing the error returned by dlopen (the result of dlerror). It's possible the library is no longer included with Big Sur, but I'm not sure (I don't feel like booting my Big Sur partition right now to find out, and googling doesn't immediately turn anything up either).
Do not try to use one of the libsqlite's from random other applications, and not just because they might not be there. That's just bad practice.