What is the aim of the procedure you describe in your post? Can I continue using the newly started set of backup's?
The basic aim is to start with a new backup set. It may be possible the current set has become corrupted ... so, yes, I would recommend that you continue to use the newly created backup set going forward. If, per chance, you experience the same issue again, we would need to look further at what may be causing it. Hopefully, that will not be the case.
Is it correct that with the "archive function" I can create a backup from the TM backup?
Yes, that is correct.
I suppose that I can start a second set of backups (manual) on an other HD, parallel to the automatic backups on the Airport TC. Is this correct?
Yes, that is also correct. Time Machine can be configured to use multiple drives for backups. Again, in my case, I use both a Time Capsule and an external drive. During the backup process, Time Machine will alternate between them ... so the process itself will not run in parallel as Time Machine can only back up to one drive at a time.
In some post I read that the issue might be that the backups are still present on my TC, but just that the dates haven't been reloaded correctly. This is the link to the post, but none of the solutions worked for me.
Pondini's Time Machine Troubleshooting articles are the "gold standard" out there and would have been where I would have sent you to next. Unfortunately, he is no longer with us and they have not been updated for awhile now. If you have already tried his suggestions, that is pretty much the most you can do at this point.