Analog to digital conversion is highly variable in quality and responds dramatically to budget. If the thing you want is ADC, you want it all spent on the ADC.
Live gear has different tradeoffs. Particularly at the low end, it's not meant to produce a signal that's pleasant to listen to alone in a silent room. As you walk up in price point, money will be spent on things you don't need, like onboard processing (your DAW is better anyway), durability for touring (overkill to sit on a desk in your house), convenience of managing chaotic high-channel-count situations (studio work is methodical and iterative), integration with proprietary low-latency networking, etc. Pristine, noise-free preamps are pretty late in the list. Different needs, different tradeoffs.
The very best digital live consoles asymptotically approach the capabilities of Pro Tools on your Macbook. Having to operate in real time adds a few zeroes to the cost.
If you really must record several mics at the same time, and can't afford enough ADC to bring in the tracks separately, then it might make sense to mix the signals upstream of the recording using whatever is at hand.
> how does the choice of live mixing console vs audio interface affect production?
1) A mixer "mixes" multiple inputs down to one output. An audio interface maintains separate channels.
2) Prosumer mixers don't have MIDI-in. That's often ok for just guitar or vocals, but add a couple of synths and the routing gets complicated fast. Hope your computer has lots of USB ports!
3) The dedicated audio interfaces have adequate DACs, no idea about the prosumer mixers.
Most people doing recording want to keep the channels separate and mix in their DAW, so they don't use mixers for recording.
In this case with the AG03, there's only one or two channels, so he's really just using the pre-amp feature in isolation.
So technically he's using the wrong device, but the AG03 is so limited it doesn't matter. :)
I have a Yamaha MG10XU 10-channel mixer, and I wouldn't use it for recording since the 10 input channels get merged into 1 stereo output channel, so there's no way to individually adjust gain or effects in the DAW, only using the physical knobs. I always use my Focusrites, Steinbergs or synth interfaces for recording.