The USB is not going to help you in this application.
Ideally what you would have is one or more of the
MiniYGDAI cards to provide a multichannel digital output from the board (In a variety of formats - Tascam, AES/EBU, ADAT Lightpipe, etc). Then you would need a digital audio interface card for the Mac with inputs capable of handling the number and type of outputs that the YGDAI cards are presenting.
Then you would control routing of the signals to the outputs through the boards software/interface. Since you are not likely to want to record every input discretely, you would do some grouping or submixes, and record, say 8 tracks or stems, each of which may have from one to many inputs.
Some sketchy examples
here
This will cost at a guess $2 K - $3 K to set up
The other way is to split the inputs as they go into the board, with a direct box or similar on each input, and take the split inputs to the computer's interface. Advantage of this is that the computer tracks are unaffected by the house mix. Disadvantage is that you will need as many inputs to the computer as there are inputs going into the main board, meaning either a ferociously expensive interface (once you get beyond 8 simultaneous analog inputs), or a second mixer to do a submix for the computer.
The third way is to dive into the documentation for the Yamaha board and see if there are submix or group outs that will allow you to take analog outs of 8 or so subs without affecting the house. Whether you can do this will vary with the design of the board.