The solution is simple. Families must choose a school prior to the official start date of their freshman year. Once they've selected a school, they're locked in. Period.
Any transfer that occurs after the official start date of their season (or first day of school for non fall sports), they are ineligible for interscholastic competition for one calendar year.
Of course there are extenuating circumstances to this rule such as hardship appeals (voted on and approved by the FHSAA in a more timely manner than in the past), families moving into the school district, etc.. Just like any system, it would have it's kinks that would need to be worked out.
It is important to note that this might not prevent "Super Teams" from forming, but what it would do is put kids in a system - academically, socially, athletically, and for some spiritually - that would allow them to grow where they were planted. Kids would then learn the merits of loyalty, commitment, overcoming adversity, and working towards a common goal with mostly the same teammates they started with their freshmen year.
Perennial powers like STA, AHP, and the like will still likely occur - and might be even stronger - because they are going to attract top prospect athletes from an early age and will matriculate through their programs over four years. For the rest of the programs, you would have the natural ebbs and flows where teams are really solid - playoff and/or potential state-championship caliber - for several years, and then they'll have a year or two where they are "down" due to possibly a lack of talent or inexperience.
I'm not sure how anyone can argue this is a not a promising thought process. I'm not saying it's the only way to even the competitive balance either.
Just a nickel's worth of thoughts...