Are you using plain ALSA, Pulseaudio and Jack, or are you using Pipewire, Pipewire-Jack and Pipewire-ALSA? (Not sure it would change much but it’s still an interesting piece of information to have)
What frontend for configuring jack you are using? qjackctl? You can try cadence or can check if killing pulseaudio (pulseaudio -k) before trying to start jack shows any difference.
Ok, I got it working with Cadence somehow (I see 0 difference with qjackctl’s connection graph but whatever, it works).
It would be perfect, if I figured out how to connect the output to the integrated sound card and the stereo system connected to it, but I guess, it easier to just connect the stereo to the external card.