I was under the impression that Dropbox stored the password because it was able to reinstate itself as an accessibility service as many times as it liked withou

I was under the impression that Dropbox stored the password because it was able to reinstate itself as an accessibility service as many times as it liked without having to ask for the admin password That is not necessary. A SUID binary owned by root runs with root's privileges. So, they only need the administrator's password once to install the SUID binaries. Afterwards, they have their own 'backdoor' to reinstate the accessibility settings, without needing an administrator password. So, when they deny storing your password, it's probably true, they don't need it. (If you are not convinced, write a small C program that executes a shell, compile it, make root the owner, set the SUID bit. You can be in a root shell without ever typing a password. This is why it is a good practice to have as few root-owned SUID binaries as possible.)