After loading a package, does R still depend on accessing the package's directory?

Yes, in several different ways.

As @M-- said in a comment, most functions are not loaded into memory when the package is loaded, they are loaded on first use. So generally you need the files to load from until everything has been used once.

The help system doesn't load much into memory when the package is loaded, it responds to help requests by going back to the file system. Vignettes are the same.

Many packages also make use of the system.file() function to find the location in the file system of particular resources. For example, a package might store images there, or fonts, or some foreign language code, and one of its functions might access the resources every time you call it.