Won’t the supported reagents leach since they are adsorbed and not bound?

It depends on how they are used. To be effective, the supported reagents such as Si-KMnO₄ should be used in a solvent where the free reagent is not soluble. By doing so, the reagent stays on the silica for which it has more affinity.

If large amount of water is present, the reagent will be completely washed off or destroyed.
For example, when working with Si-KMnO₄, an organic solvent such as cyclohexane should be used (see SynLett 2001, 10, 1555-1556).

Unbound PCC and PDC are almost always used as a suspension in DCM, adsorbing it onto silica facilitates its removal. The reagent cannot technically leach since it is not soluble.