Until someone tries, I can only imagine how the thingy performs...
As compared with a phase transfer catalyst, the disks avoid one added compound which I suppose sticks at the interface of the reactants to stabilize an emulsion, hence resides where the reaction takes place.
Compared with a stirrer, the disks work by letting one reactant (or both) adhere on them to form a film, so they don't need a stable emulsion. They also avoid long and brutal stirring of two badly miscible reactants, which would heat the mix.
The disks can also be a catalyst or a reactant, like Zn, Zn-Cu...