I'm not sure "broadly popular" is the right phrase. I think "not actively opposed" is more accurate: most people accepted Vichy, but weren't enthusiastic about it.
Pétain was popular and seen as a savior. And even after the war, most people (including Resistance veterans) believed, or wanted to believe, that Pétain's collaboration with Germany had merely been a ruse to protect France from the occupier (while preparing for resistance). Ironically, most German officers and party officials believed it too during the war (which gave it more credence later). BTW, this idea that Petain only pretended to collaborate is a lie.
Pétain was popular as a figure. His prime ministers, however, were not (and took the blame for everything). And Vichy as a regime had limited popularity.
The bulk of conservatives, right wingers and fascists were happy with Vichy's internal policies (giving prestige and some privileges to the Church, anti-communism, forbidding strikes, antisemitism...).
French fascists were
happy with collaboration and wanted France to fully participate in the New Europe (and fight Judeo-Bolshevism) with fervor, but those people were a small minority.
Most right-wingers didn't like to see France as a German vassal (as they were patriots) but at least, Hitler and the Vichy regime opposed the Commies... and there was "no choice" anyway since Germany had won in '40, so France should curry favor with Berlin. However distateful it could be.
Those right wingers either collaborated with Germans, or supported Vichy's internal policies while being apathetic towards Germans, or
supported Vichy's internal policies while resisting German occupation. (It needed some interesting mental gymnastics).
If Germans had accepted to turn France into a true partner, all those people would have supported Vichy and the collaboration even harder (and with less reluctance and ambiguity).