Do they? The only instance when they did keep someone from leaving was Texas to the PAC and I’m not sure Texas really wanted to go.
Will the state politicians be willing to tick off the UNC and UVA alumni/fans by holding them together? They could, but will they?
No they will not, unless they don't care about the hundreds of $millions they would be flushing down the toilet.
The Virginia governor famously twisted Virginia’s arm to vote VT into the ACC. A good example of state government intervention.
Sadly, the Virginia governor and the Carolina board of trustees have no leverage with the SEC or Big10. They can make all the noise they want, no one will care.