It's feeding a radio control servo - the position is defines by the pulse duration.
(Usually 1 - 2mS range, with 1.5mS being centre of travel).
They are supposed to have a continuous pulse train with ~20mS repetition rate, though. The behaviour cannot be guaranteed if the pulses stop, as they would do with that circuit.
I think the OP is controlling a model servo which requires pulses between 1 and 2 mS and has all the feedback built in.
The circuit appears to produce 1 or 2mS pulses depending on which button is pressed. This should be producing pulses every 20mS and some servos don't like if the timebase is different.
There's a similar thread from a long time ago here.
Mike.
P.S. sorry JR (or RJ) typed this about 6 hours ago but didn't post it.
Can't help with Falstad - don't even know what it is.
I'd approach this as needing a pulse every 20mS - a 555 setup to produce 50Hz will do this - positive edge every 20mS.
Use the pulse to trigger another 555 to output a 1mS pulse.
Have a push button that changes the period of the second 555 to 2mS.