Make execute_and_log_request a method of Client.

Make execute_and_log_request a method of Client.

All the calls to it happen inside methods of Client, so there's no
reason this is difficult. The only reason it was a separate function
in the first place was so that the old version of login.rs could use
it, which did login outside the full TUI via a separate network
interaction module. But now all the functions of that module have been
moved into Client, execute_and_log_request is fine as it is.

It also doesn't need to be pub any more.