I think it depends of the gains of the transistors whether the capacitor is needed or not.
But to be on the safe side, it deserves a place in the circuit 😛
Aha, your solution to the led problem is a good find, but not the ideal solution !
Indeed, when shunting the output with a resistor will make it work. But the current draw will increase.
Lets assume that we would like to save our battery thus using a led. So, the first design (using your better setup) should be modified by changing 2 components.
The question remains, which ones ?