Even though the DT will play whatever notes come through each track’s assigned MIDI channel, it uses the AUTO channel to live record MIDI data for whatever track is currently selected. It’s odd, and in your use case annoying, because for you to record your drum rack this way, you’d have to go through it track by track and change each corresponding part in your Drum Rack to hit the AUTO channel.
There is an alternative, which is that the DT also always lets you play and record its tracks using the first 8 notes starting from C0, regardless of the current track selected. So if you can assign each of your Drum Rack parts to send the right note (like, your kick is on track 1, have the kick in Ableton send C0, snare on 2, send C#0, etc) then it should record the way you want it to.