With all credits and many thanks to Jacob Peattie in the comment above.
After Jacob’s remark that the title is used for e.g. the title element and by some plugins, I followed his advice to create a new field because he made me realize that the front page tagline needed somehow to be seperated from the other places where the description is being used.
So I created a custom field in the WP settings, a function to (only) allow HTML for HTML anchors (All other code keeps decoded to entities) and a function to parse the alternative tagline.
Then the new function needed to be implemented in the theme, so I created a child theme with the new alternative tagline in header.php
So not a simple plugin as I hoped for, but a child theme with all good reasons 🙂 And it works like a charm.
Thanks Jacob!