Well, there's a possibility that "choice vouchers to help public kids go where they want" could happen. Are you familiar with the Backpack Bill? It is a piece of legislation being workshopped and pushed that is standing front-and-center of the 'school choice' movement in Ohio today.
Regardless... I think the "closed enrollment schools would flip out" belief is, well, flimsy.
What are reasons that schools are closed enrollment in the first place? Primarily capacity reasons, I suspect, but also likely a firm belief that you ought to live in the district to attend. The majority of these places are considerably wealthy. On the surface they might care a little with funds leaving the district, but the majority of these places have pretty competitive academics/extracurriculars, superb facilities and enough "draw" to attract -- and retain -- its families. In other words, they likely don't stand much to lose in the student game in that hypothetical.
Also, they probably don't give two craps about how it may make a small dent in athletics.
Here's a list of closed enrollment school districts in Ohio...
Forest Hills (Winton Woods)
Hamilton Local (which is Hamilton Township)
Highland Local (Medina County)
Jackson Local (the Jackson that is in Stark County)
Lake Local (Stark County)
Lakewood (Cuyahoga County)
Middle Bass (???)
North Bass (Kelley's Island?)
Perry Local (Lake County)
Plain Local (GlenOak)
South Euclid-Lyndhurst (Brush)
Southwest Licking (Watkins Memorial)
Southwestern City Schools
Springfield (Lucas County)
Springfield (Mahoning County)