Turns out that ScanGear MP 3.70 works perfectly with my Canon TS5050. After unpacking the archive and
dpkg -i scangearmp2_3.70-1_amd64.deb, I ran
scangearmp2, got simple GTK-based GUI, the TS5050 was recognized, and that was it.
I have no idea if it integrates in desktop GUIs, since when I'm under Debian GNU/Linux these days, I am happily using i3 and terminals.
It would be really nice if Canon listed any software for Linux on their official support page for Canon TS5050, but they don't. There are no Linux "bundles" listed (whew!), but this also means there is no software offered at all. Once you choose Linux, you get random offers for Windows and Mac software — but that's just generic ads displayed under the 'no results' area.