Thanks @duncanc, that helped me solve my dilemma!
It turned out that it was actually working correctly, but since I don’t have a mail server configured on my localhost, phplist was responding:
Sorry, sending the message to request your confirmation failed, please click "Reload" to try again. If it still does not work, please contact the administrator.
When I uploaded the code to my server it worked as expected.
So, for anyone else having trouble with this:
You need a working mail server in your development environment to test this.
(Which I should have remembered from reading the phplist development guide…)