I think its Favorit 135 / 1.3 43 (https://en.wikipedia.org/wiki/Å koda_Favorit) It's from 1994.
It's definitely not transmission (I can put it in neutral with same results). And it doesn't make sense to me to be fuel related since it just started in the middle of road. I'd say its 90% carburetor but I would need to know what specifically could be the culprit. Like I said, engine works "fine" as long as you are constantly accelerating - so to drive it you literally have to tap the gas pedal constantly, if you stop in any position - doesn't matter if it's pressed to the fullest - it just loses all the power and dies. It's... weird.
I used to have some Pierburg manuals but can't find them online anymore. Anyway, IIRC there is a separate channel(?) for extra fuel if you are accelerating so I'm thinking maybe only that works and main(?) thing somehow doesn't?