Steve, thank you for your feedback.
My case is a Demo Course which must be accessible for everyone once registered/logged-in. Why do I need all the users to be assigned to a group - because - it's a kind of integration specific, the integration with an external system (business simulation) where users should be assigned to a group which is replicated from Moodle.
Of course I might have have used the group enrolment key, just put it in a welcome text and that's it. Of course I can re-develop the integration stuff )) Just wondering if there is any fast and easy solution that might help to bypass any extra clicks for customers.
Meanwhile I came across this - https://moodle.org/plugins/view/enrol_autoenrol, it almost works as needed if I use the "User Filtering" options...