Ohio ASA determines when States are played. The Commissioner is Warren Jones. The situation with respect to timing is better than it has been in past years, when it was sometimes held even later than June 22.
A better solution to the problem that you identify, in my humble opinion, would be to allow those teams that have already qualified to partiticipate in States. Ohio ASA has the authority to do this (and indeed, in a true National Qualifier as that term is defined in the ASA Code, the Code specifically states that existing berth holders may participate without forfeiting their existing berth) but has chosen not to. If an existing berth holder were to win or place at ASA States, the ASA States berth to ASA/USA would not evaporate, but rather would be passed down according to very specific tie-breaking rules. The only remaining argument that I can see against allowing existing berth holders to play in ASA States would be that some good teams might be eliminated from the tournament by a team that already has a berth.
The problem with excluding existing berth holders from ASA States becomes most important at the 16u age group, as this State tournament draws a lot of college coaches.
BTW, I would be surprised if there are four ASA/USA qualifiers at 12U prior to States. I am aware there is Dayton Metro and perhaps there is a Region 9 open and/or closed qualifier (I just haven't been following the dates of the strictly 12u tournaments). Strongsville, which is held after States, is the only remaining qualifier unless you live in a geographical area that is eligible for the Cincy Metro qualifier.
Finally, especially at the younger ages, I don't think a team should pass up early tournaments offering berths on the fear that the team won't be allowed to play States. We are not going to be able to play States at 16u this year, and that stings, but I can't imagine that we would forfeit our existing berth or that, had we not earned the berth last summer, we would have skipped the ASA/USA qualifiers that precede ASA States this summer.