This is an inadequate answer due to a lack of time and the fact that I am not expert on French interwar politics.

However, it must be understood that Petain's career was one of crisis management: he took over at Verdun when defeat looked likely, introducing new tactics and a humanitarian approach that sought to alleviate the suffering of front-line troops. His methods contributed to the victory there.

In 1917 he took over control of the French Army from the confident and charismatic Robert Nivelle, whose grand offensive had fallen far short of his big claims for it, leading to many units of the French army refusing to obey orders. Seen as a mutiny, the rebellion was more nuanced than that but led to Petain having to take control and manage the situation to keep the French Army fighting. Again he succeeded and although he was seen as very pessimistic, he was respected for his humanity and for his rejection of the 'attack at all costs' mentality that preoccupied most French high army commanders.

For these reasons, it was quite natural to look to Petain after the German army swept through France in 1940.