In classical Marxist terms, a Communist revolution needs both the subjective and objective conditions to thrive, in other words not just a revolutionary ideology and a faction to promote it, but the class basis for its ideological maintenance and the economic means to deliver socialism and ultimately communism.

In these terms the Third Estate's more radical element could have established a communist party had it possessed the ideological wherewithal, but it wouldn't have got very far because the landless working proletariat was as yet a minority and France was insufficiently developed to progress to a self-sufficient socialist economy: in the event it failed even to sustain even the bourgeois Republic or the subsequent Empire against its external foes.

Ultimately it's down to the development of a propertyless working class with "nothing to lose" and everything to gain from the socialisation of property: France in 1793 remained a predominantly peasant nation, perhaps 18% urban, little more industrial, in which the material interest of most lay in adequate land for each rather then the abolition of private property. That was more than enough for most of Europe's rulers.