it appears the cage can only hold one crystal at a time
It might be more like a single terminal for a "micro-turbolift" system that delivers the crystals deep into the bowels of the lethally radiating reactor machinery - but with multiple "turbolifts" to receive and deliver the crystals, either to their points of application, or then to waiting positions.
The little "lifts" could then accept either macroscopic natural crystals in those holding frames, or arrays of microscopic (synthetic?) crystals in the ping-pong paddle frames seen in "Alternative Factor".
This would lift most restrictions on engine room relative geometry, as the "turboshafts" could zigzag anywhere. The pedestal Spock manipulates in ST2:TWoK to bring main energizers back on line would be the more modern equivalent of this dumbwaiter system, similarly delivering to some underfloor reaction focus... If need be, to the one seen in ST6:TUC and featuring a TNG-style dilithium chamber!