id say it's your accelerator pump.
when carbies are old the diaphram cracks and no longer squirts fuel. Have a look with the air cleanes off, look down a barrel and see if you get a good squirt like a water pistol of fuel when you move from 0-100% throttle. (with the car off of course).
This would also explan the stutter when you put your boot into it.
This also expalins the car not starting since you need to pump in a bit of fuel to get it to kick over usually. and no accelerator pump = no start.
You may just be able to buy a new diapram for the pump but i'm not too sure. you might have to buy a whole kit. Not sure how much though.