I agree with both of you.
The two-man system is inadequate for HS soccer in TN.
As much as I hear coaches and ADs bemoan the added cost of the third official, the root of the problem is NOT financial - it is the lack of officials. I'm sure most assignors would agree - there are simply not enough available officials to cover the contests.
This also isn't about quality - that is an entirely different subject. But, IMHO, the referee + AR + AR system is a much better training ground for referees, IF the center has some experience and is willing to do some mentoring of his/her ARs. ARs willing to observe and learn can work their way into the center position over time, but if an inexperienced official (or two!) shows up to find they are running a 2-man system everyone suffers.
Too many refs go directly from doing U12 rec games to HS without adequate seasoning. The game is different, the coaches are different, the rules are different (and let's not even talk about the hokey HS signals!) and the speed and intensity of play are far different.
All of this hinges on training, good assigning, personal conditioning, etc. There are problems there as well. However, a larger pool of referess would again alleviate the problems.
OBTW, if we were blessed with a wealth of competent referees, the 3-whistle system contained in the NFHS handbook could be implemented. This would do the most for fair, clean, safe play - players hate it, particularly less talented male players, because there are three people observing the action, any one of which can stop play, call fouls and issue cards at any time.