Hmm.
A bunch of companies (Allen & Heath QU16, Presonus Studiolive 16, etc.) make 16 channel digital mixers w/ lots of inputs that also offer DAW control. Certainly at the higher end it's easier to find (to continue w/ Presonus example, there's the Studiolive 32sc)
Many other companies (Mackie MCU) make 16+ channel controllers which work w/ separate interfaces.
I don't know offhand if anyone makes a cheap 16+ channel portable mixer which ALSO works as DAW control AND which has built in recording, though I would check the Zoom L-20 to be sure, and would call somewhere like Sweetwater.