LED Bulbs not Starting Properly
Halogens are incandescent and do not require a "starter". Those are for fluorescent lamps.

I am guessing the LED's do not put enough of a load on the secondary (12v) side of the circuit to initiate a steady current draw. Some power supply output circuits are electronic and not just a straight transformer.  You would need a ballast resistor added to the circuit in parallel with the lamps to get all three LED's to work. But then you have current flowing through the resistor all the time, defeating the purpose of switching to LED's to save electricity.

You should get a fixture specifically designed to use LED's.

The specifics of this question are probably better answered by an electrical engineer so you would need the question moved to the electronic store section if you want an in depth response.