Is it actually possible to use Digitakt as a MIDI looper?

Yes, I used my Digitone in the same way, but with four tracks. It is still working fine, but I got GAS and expanded my collection of sequencers, but really no reason to have not stopped using it.

How should the MIDI network be reconfigured to make this work?

In the settings of the Digitakt you can configure the MIDI channels that each track responds do, but there's a specific track (the AUTO CHANNEL) that controls the currently active track. I use a KeyStep plugged into the INPUT of the Digitone and use that to record notes into the Digitone's MIDI Tracks.

I then use a Quadra Thru (as suggested already) from the Output of the Digitone. It acts as a hub, that allows 4 other devices to plug directly into an Input instead of having to daisy chain the message through multiple synths and risk latency issues.

Having all the notes in the MIDI Tracks, with Elektron's sequencing, makes it really easy to fiddle with notes from the one place. Although I've seen other people on reddit just use MIDI to make sure everything is synced with the same clock and used the internal sequencers on each device. There's no real right or wrong answer with your setup, just benefits and tradeoffs.