> There is no retirement “fund” in the way that word implies
well, France really has 2 systems (to simplify, it's really more like 42+), the "public" one (i.e. civil servants) and the "private" one (i.e. anyone employed by an employer other than the state).
The "public" retirement is based on the assumption that the state guarantees its liability for future pensions but doesn't actually put it on the books (if it did real debt would be another 2,000 billions euros), while the "private" retirement does have a "fund" (really several of them, one per sector of the economy) but it's revolving money put in by the current workforce but paying immediately current pensioners. This second system can in theory have excess money (but every time it does, a law is passed to extract some more money out of it to the benefit of the first!).
Part of the issue in France is the existence of these 2 systems and the very fact that the state allows itself to do its accounting differently (in a way that unlawful for anybody else). If at least the state played by the same rules... (of course then it would have to recognize this extra debt of 2,000 billions euros, and that would disqualify the country from the eurozone!).
But in short, yes it is very much the government's problem. They (since 1945) have "organized" pensions in a way that makes them a time bomb both in its accounting, and in its unjust rules (rules vary a lot per sector of the economy).