I haven't read all of it so I'll comment on the part I did and remember. If I'm off, you can steer me right Fish.
First and foremost its not a simple tax, even though its advertised as one. It is much simpler then the present mess. Depending on how its written, it could very quickly become just as complicated as the present tax. I don't think anyone would believe it can eliminate the IRS, thats simply doesn't compute.
I haven't gone far enough to know what their suggested rate is, and how thats going to replace the present rate, which we admittedly can't compute.

There would be nothing simpler than keeping X number of dollars out of your "checks", no matter where it came from and with a consistent sliding scale with a flat tax for income beyond reasonable earnings.
The individual, at the end of the year, could the send a postcard in, top line = how much you made, second line = how much tax you paid = third = how much you should have paid, fourth line = any refund you might have coming.
Each individual could make sure, quarterly, that his rate is consistent with his income, and it could be done in a couple of minutes.