If I understand correctly, you want to send the audio of the separate channels to an Arduino, which then sends it over USB to be recorded in Ableton.
Is that what you meant?
First off, Arduino's don't have any hardware to record audio. You'll need external I²S ADCs for that. The next problem is the USB connection. Most basic Arduino cannot support audio over USB, and the more advanced ones that do have the hardware to support it, don't have software that supports it.
One exception is Teensy, which supports at least stereo audio over USB. More than 2 channels should be possible, but you'll probably have to dig deep into the USB code, which isn't something I'd recommend to a beginning (or even intermediate) programmer.
I think the Teensy 3.x boards support up to 16 audio inputs using TDM. Again, configuring these is not going to be easy, although the Audio library seems to support TDM.
If you're super passionate about learning about the details of the USB protocol, DMA, real-time audio, etc. and if you have the necessary programming background, by all means, go for it.
If you just want to record audio, get a USB audio interface, and plug it into the channel inserts of your mixer. If it doesn't have inserts, you could use some auxiliary busses to record specific instruments.
There are some mixers with multitrack recorders built-in. A couple of years ago, I bought a Behringer UFX1604 for around €600. It records and mixes fine, but some of the mute/solo buttons are starting to get stuck. Other mixers with multitrack recorders built-in were much more expensive back then, so I think it was a good deal. I have no idea what the market looks like now, though.