An occasional drop-out like yours can be due to a number of factors. Given it is always from a stop, and even then it's intermittent, it must be something that happens whilst the engine idles, and then when you hit the gas it purges the problem (after a brief
hiccup). Let's look at a few things you mentioned...
Fuel pump... well, it's clearly pumping OK, because you have no issues 'at speed', where you'd see any pump supply issue most exacerbated. So, I think we can rule that out.
Second, ZS (and S.U.) carbs have no accelerator pump, per se, but the oil in the dashpot acts as an ersatz pump... it slows the rise of the air valve (which is lifted by engine vacuum when you press the throttle down), allowing the intake runner vacuum to momentarily
increase over the metering jet orifice, thereby creating a momentarily greater mixture when you depress the throttle. Lack of oil in the dashpot (or a very thin/light oil), gives you minimal 'accelerator pump' and conversely the proper oil (normal engine oil)
gives you the proper 'accelerator pump'. So... since this is intermittent, I doubt it's the dashpot oil - if you lacked dashpot oil it would give a consistently poorer off-the-line performance (NB: you should be checking that oil as often as you check the
engine oil, which one presumes this would be every few hundred miles or so).
So... having ruled out those two, what else may be at fault? Following the fuel issues theory, one might think it was a carburettor diaphragm, but again, this would virtually always cause a consistently poor performance, not an intermittent issue such as you
described. It's not a vacuum leak, either, for the same reason.
However, there are two other possible fuel issues I can think of...
First, a bit of grit may have got down into your fuel jet orifice, and when idling it may be get itself into a spot where it clogs things things a bit, until the vacuum over the orifice dislodges it. I doubt this is the case, however, but it bears checking
for dirt in the bowl.
Second, you may have a bit of water in your fuel tank, and as the car idles this accumulates in the bottom of the fuel bowl... when you hit the throttle it pulls in the glob of water and coughs, then starts pulling good fuel and runs just fine. I think this
may be entirely likely. To test this, disconnect the fuel line going to the carburettor and place it into a 1-gallon container (a dry milk jug works well). Have a partner turn the key on and allow the pump to nearly fill the jug. Turn off the key, and reconnect
the fuel line. Seal the jug and allow it to sit overnight, then check the bottom of the jug for tiny droplets of water... if you see even one, you've located your problem.
As to it being electrical, that's also possible. However, given it's intermittent, and only happens once at start off, I'd give low odds to it being an electrical issue. That being said, the first place I'd check is the condenser, since as they get hot they
can begin to perform poorly, and the current batch of available condensers are total crap.
My '78 Roadster with ZS carb. is experiencing an occasional hesitation in the engine. Almost like it stops for a faction of a second. It happens just once each episode and mostly happens when pulling out from a stop. It doesn't happen every time but
maybe every 3rd or 4th stop. And the engine runs perfectly at all other times.
The fuel pump is pretty new. This has been happening for several months and a few hundred miles. It's been refueled several times with Sunoco 93 since it started. Accelerator pump? Something else?
Thanks for any ideas.