ukn_unknown
So finally I did pretty extensive experiments to figure out the show-stoppers in my setup.
I won't post any logs or detailed protocols of my experiment, but rather my perceptions from the experiments:
It doesn't matter whether I use vaapi based codec profiles or software profiles, when I don't enable hardware accelerated decoding but have deinterlacing enabled (SW Deinterlace), the resulting stream has extreme stuttering and is almost unusable. If I disable deinterlacing at all, it works fine, if I enable it with HW decoding, it works fine.
The assumption that "unconstrained" causes any problems was totally wrong and maybe overlayd by this problem above. I also got some libav errors when I selected some special encoder settings that weren't supported by my VAAPI, but the fact that this causes errors is indeed totally clear to me.
So finally it boils down to the following settings:

Where I either have to enable Hardware acceleration (for decoding) or completely disable Deinterlace.
I assume that some TVHeadend update changed the default settings and thus my profiles were kept but with hardware accelerated decoding disabled so they seemed to stop working, but indeed it was only the wrong setting.
Another question that arose during my experiments, and maybe sb. here can answer this, was the difference between the bultin/av-lib containers when I create a transcoding profile? When to use which of them?