No it doesn't support that, but systemd-socket-proxyd does. So my Systemd service exec line looks like this:

ExecStart = /path/to/systemd/lib/systemd/systemd-socket-proxyd --exit-idle-time=20min /path/to/jellyfin.socket;

https://github.com/attilaolah/dotfiles/blob/144d1711755d7a9e...

To be clear, it will shut down Jellyfin after 20 minutes of inactivity on the socket. It may as well be that Jellyfin is doing some background work like re-indexing videos, so it is not perfect, but it is the next best thing until Jellyfin supports first-class socket activation.