MC34063 inverter stops switchng at low current and burns out through inductor
Your schematic shows no input capacitors. What are you using there?

And you have a "breadboard" with what parasitic inductances? Post a photo?

Between inductances and capacitances, you have many opportunities for ringing.

! amp switched in 10 nanoSeconds into 100 nanoHenries (4" of wire) will generate
(wait for it)

V = L * dI/dT

V = 100nanHenry * 1 amp / 10 nanoSeconds = 10 volts