Encountered the same issue as OP in bookworm. I was not using the deb
package before, so not sure if the bug existed in previous releases.
Had to do the following steps to get the browser to launch after
completing the automatic download using "Launch Tor Browser" and
completing the download:
$ cd ~/.local/share/torbrowser/tbb/x86_64/tor-browser
$ ./start-tor-browser.desktop --register-app
This creates a new entry in the menu "Tor Browser" that can be used to
launch tor.
If the "Launch Tor Browser" is selected, it again downloads the browser
but does not launch.
Note that the package should also have
Depends: desktop-file-utils
since the update-desktop-database command is required by the
start-tor-browser.desktop script.
