Will,
Once again, I doubt you want to be bothered by the facts, but here you go:
Boys Sports
Baseball-- Current 2A and 4A Champs. Private schools have won both of these divisions for SEVEN years in a row. Since there are no 6A schools and there is no 1A baseball, private schools have won at least 50% of the divisions they compete in for the last seven years. Some years it's been 75%.
Football--Current 4A and 3A Champs along with 4A runner up (Hayden). STA and Carroll also were semifinalist teams in 5A. I'm not sure to what extent there are private schools in the 8 man divisions. I don't think any compete in 8man II. Regardless, it's safe to say that 2 champs is statistically more than to be expected.
Basketball--5A Champs, 3A Champs and 2A state champs. Also 2A runner up and 2A third place teams. State champs in 60% of the divisions they compete in.
Wrestling--You are not very good wrestlers. No current team champions. However, since 2005 there have been 5 private teams champions out of the 24 total team champions which is still 21%. Not as bad as the other major sports, but still more than you would expect from 8% of the schools.
Track--Apparently you guys are slow. You haven't won a team title since Carroll in 05. It's probably attributable to poor coaching. The public schools have really good coaches, they make the kids run really fast.
There you go. I think if you look at it objectively you must admit there is disparity. I'm not certain, but I would guess it is the same or worse in the girls division as I would replace wrestling with volleyball where private schools won 3 of 5 classifications.