The NRST due on that item would be $23, which is 29.87% of $77 (as well as 23% of $77 + $23). The state sales tax on that item would be $3.85 (5% of $77, or 4.76% of $77 + $3.85). The final item price would be $103.85, of which combined federal and state taxes would be $26.85.
As for why tax-exclusive forms definately make the implementation easier, note that $26.85 is 34.87% of $77, and that the 29.87% NRST and 5% state sales tax add up to 34.87%. On the other hand, adding the inclusive rates (23% + 4.76% = 27.76%) doesn't work, because the actual tax-inclusive rate is 25.85% ($26.95/$103.85 = .2585).
When I went to school the $103.85 would be "the gross payment"...
So using your new math, which part of your calculation would you describe as "the gross payment"?