Yes, please

A possible consequence of this would be that a course can not have two simultaneous livestreams. This might be an issue for courses like the FVV, where English and German events are streamed at the same time.

Perhaps on the course > settings page there could be a tab much like the "Course Admins" tab where a lecturer can add a streaming key (secret) and name it. When creating a new lecture, these keys can be chosen from a drop-down list or an entirely new one can be generated. This would mean a lecturer could keep the configuration of their streaming software for most lectures while still allowing lecturers to use multiple keys for e.g. concurrent streams.

If this seems reasonable, I would like to work on this.

I see now that the above approach would require a change to the db structure, which is quite a hassle. Instead I thought that perhaps per default when creating a new stream it would use a course wide key and when necessary a new individual one could be created in the manner they currently are.
To prevent having to make a change to the db structure one could hash a secret and unique property of the course and use that as the stream key.
I noticed there is a field called Token in the course model which doesn't seem to be used anywhere. This would not even have to be hashed as it is generated (when it is generated) in the same manner stream keys currently are. Are there future plans for this Token field? Can I use it for this?