ok, it seems, that I found kind of a solution for this.
I edited the file: moodle/enrol/self/lib.php
This contains a function called enrol_page_hook.
It seems, that - among other things - this function checks if a form was submitted or not.
So I tried to find the if-statements that made the difference and change them to if(true).
I changed this:
// if ($instance->id == $instanceid) {
if (true) {
// if ($data = $form->get_data()) {
if (true) {
To be honest, I don't completly understand what those if-statements are doing, but if I force the script to enter this block, a user is enrolled to a course without having to click the "enrol me"-button.
I guess it will have a few other side effects which I haven't discovered yet
Maybe somebody can help me with the fine tuning?