Of course it's never that simple, but it is essentially what happened.
When I hear about Italy switching side, it's like the knowledge about the country splitting in two and going in a civil war with the north supporting the Axis and the south supporting the Allies never happened. Even though it's a civil war that lasted almost to the end of WW2.
Your point would be that Vichy France is the equivalent of the Italian Socialist Republic, but it's not quite that way even though the latter is way more popular

Vichy France was the new French government, it had its original flag, the army, and the economic resources of the nation.
The Free French Forces, the ones with this flag

Were the forces lead by De-Gaulle, but it was a resistance group, mostly formed by ex-military.

If it's common knowledge among simpletons that Italy switched side, then it would be appropriate to know that France did so as well. The actual French in power of the government.
Pétain and Badoglio essentially did the same thing:
Sign a pact with the ones who invaded their country, and switched to their side, let it be the Axis or the Allies, the circumstances were the same

man history has so many little details