On a personal basis, I live in Orange County. I don't have the option not to pay the part of my taxes that pertains to schools even though I don't have children in the school system.
The entire purpose of the RCID, and, for all intents and purposes, the City of Lake Buena Vista, is for Disney to pay taxes to fund infrastructure, maintenance, and operations that specifically benefit themselves. The Reedy Creek Fire Department is funded entirely by the taxes that Disney pays and services only the area that Disney owns. The same goes for roads, water, and sewer utilities.
It's kind of like if you were able to form your own school district that covered just your own house, and then voted for yourself, set your own tax rate, and paid yourself to run your own school. The corollary would be, like in your situation where you don't have children in the school system, you could choose what services to provide and how much to spend, therefore controlling how much you pay in taxes to that entity.
In Disney's situation, if the new district, controlled by state appointees, chooses to no longer fund Disney infrastructure, then they'll no longer collect taxes to pay for it. If they no longer fund the fire department, they won't collect taxes to pay for it.
If the appraisal on Disney property increases dramatically, that doesn't change the amount of money required to pay for roads or fire fighters. Disney will still need to pay the same amount of money to fund those services. If the property is worth $100 and the district needs $2, then the rate is 2%. If the property is worth twice; $200, the district still needs $2, so the rate drops to 1%. These districts account for 60% of Disney's tax burden, so messing with appraisals won't change the amount Disney pays to those districts because they already control expenses.
The other 40% of their tax burden could be impacted if appraisals increase, but how big of an impact could that really make? What's the likely increase in value; could it double, or would it be more likely to increase 10%? It cold certainly cause an increase in taxes, but not a huge change. Between the likelihood of a change, and the impact of a change, it's not something I'd waste energy worrying about.