CTL:
The system you describe might work, but there are drawbacks. I know virtually nothing about your state. But in NY, there is a get law. This requires the obstacles for remarriage be removed prior to the civil divorce. And the typical beis din requires a resolution of the affairs by agreement or by beis din arbitration prior to the get. This is not mandated by law or halacha, but one would need to have his head examined if proceeding to the get without an agreement.
Any beis din with a moral compass will inquire whether there had been effort to reconcile before proceeding with the get. There is just variability how much of an issue they make of this. And one side will often prevent the get from proceeding quickly with the array of delay tactics, among which is the “I prefer shalom bayis – let’s go for therapy”.
But to be clear, there are instances in which a couple makes a mutual decision to part ways, reaches agreement on all issues with minimal fanfare, and seeks a “quick get”. And my experience suggests there are several batei din around that accommodate that quite well. There is also a phenomenon often called a “get beis din” which is convened to accomplish only the get itself. It avoids all the litigation that keeps many other batei din busy. This tends to be the sofer who serves as a dayan, and two others who join in to complete the beis din of 3.