Impossible : it was built on a coup against a representative regime.
A parisian mob did upset the regime and the elected national assembly.
Then the group of extremists that had the support of this mob did organize elections in a context of terror where only 10% of the electors voted : they were of course in majority the ones that supported the coup of this activist minority.
Then, the new government of the new republic had to enforce a politic of terror to crush opponents, since it actually was very far from having the support of a majority of the population.
Remember that the french revolution in these years was the model on which Lenin conceived his way of taking and keeping power.
The first french republic becoming a full democracy would mean the "republicans" losing the free elections.