How to use an EntityClass

This is more of a workaround. If you're willing and able to preload the data, you could create your own custom Entity type with EntityStore and treat it as if it were a built in.

Your data:

bport = Entity["City", {"Brockport", "NewYork", "UnitedStates"}]["Position"];
ps = EntityClass["PublicSchool", {"USState", "NewYork"}];

The nearby schools, per Jose's comment above:

ents = GeoNearest[ps, bport, 100];

Create the custom entity. For brevity I omit custom labels, properties, classes, etc.

store = EntityStore[
  "NearbyPublicSchools" -> <|
    "Entities" -> EntityValue[ents, "EntityPropertyAssociation"]
  |>
];

EntityRegister[store];

Query as if it were a built in type. For example:

largefaculty = EntityClass["NearbyPublicSchools", 
  {"TotalTeachers" -> GreaterThan[Quantity[50, "People"]]}];

largefaculty["EntityCount"]

28

GeoListPlot[largefaculty]

---

Here's another workaround, but unfortunately your example returns Missing["QueryTooSlow"].

EntityClass["PublicSchool", 
  {"Entity" -> ps, "TotalTeachers" -> GreaterThan[Quantity[200, "People"]]}
]["EntityCount"]

Missing["QueryTooSlow", {"PublicSchool", {EntityProperty["PublicSchool", "Entity"] -> EntityClass["PublicSchool", {"USState", "NewYork"}]}}]

Here's a smaller example:

EntityClass["Lattice", 
  {"Entity" -> EntityClass["Lattice", "Even"], "Determinant" -> GreaterThan[10]}
]["EntityCount"]

6