Max Sinister said:
The POD is either to the end of the first French Revolution War (1792-97) or during the time between the first two wars (1797-99). So or so, Napoleon is killed - a stray shot, assassins because some people found him too dangerous, catching a disease in Egypt, something like that.
Could the young French republic survive under the Directory?
Under the directory? No. Corruption was rife and the people have had enough. There were already a number of conspiracies, some of which, like Hoche's, came within a hair of suceeding. The time had come for a Man on a White Horse.
If Napoleon is dead, the conspirators which tried to use him as a figurehead ( that plan didn't quite work )will turn to the next most popular sucessfull general. That being Massena at this point.
Massena was greedy, but also really a republican. He will take power to save the Revolution ( and also to enrich himself ) but he will never proclaim himself emperor. France will stay a consulate, maybe even with a consul for life - but I doubt it -, republic.
Max Sinister said:
Some sources say that France was practically broke at that time... and when the 2nd Coalition War breaks out, do they have a realistic chance to survive without Napoleon's genius?
Most definitely yes. Most of the genius of Napoleon was in the Grande Armée. From the Veteran troops to the Marshals, from the generals to the arty corp inherited from the royal army. Those are still here. Not to mention Massena was not bad himself when it came to war ( just ask Wellington ).The 2nd coalition is going to get crushed on land, no question about it.
However, a point to think about is whether the war will rekindle with England as it did OTL. Massena will have no problem evacuating North Italy; it wasn't HIS conquests, so the war party in England has one less pretext. The war could be postponed for a cvouple years, or even not include Uk in the coalition; Either would be very good for France.
Finally, Massena would not put his brother on the throne of Spain and would avoid attack Russia is he could. He also didn't have Napoleon's ambition.
Most likely the french revolutionary wars stop around 1808, with the map of Western Europe redrawn and France stays a Consulate republic untill nowadays.