The point of an API is that its behaviour is predictable regardless of the implementation. Tying all the system mail clients to a specific implementation of that API means that they'll all mysteriously break if the admin decides to replace sendmail with, say, exim. So, instead, applications would have to use some other interface than /usr/bin/sendmail, which is where we started in the first place.
"I remain unconvinced that the fact that the configuration requires admin privileges is a real problem."
The modern world considers mail to be per-user configuration, not per-system. If you're maintaining a lab with a bunch of users who want to be able to use Thunderbird with their gmail accounts, are you really suggesting that the appropriate thing to do is for the admin to add their gmail credentials by hand?