Ummm, no.
Paul Prudhomme decidedly did NOT invent brown roux gumbo.
"It was only toward the end of the 19th century that the ingredient that's perhaps most closely associated with gumbo today made its way into the pot: roux. A staple of French cuisine, it's a blend of flour and fat (usually oil for gumbo) cooked together, used as a thickening agent and—particularly in its darker Creole and Cajun variants—as a flavoring agent, too.
Adding roux to gumbo was definitely a Louisiana innovation, and it seems to have been used first with oyster gumbos. In July 1880, after the
New Orleans City Item ran an article on gumbo, they received a letter from "a lady who is a very competent authority in such matters" instructing them on the "genuine Creole method of making gumbo." She provided instructions for a standard chicken and okra gumbo with no roux, then followed it with a recipe for gumbo filé with oysters. It begins "make a roux and brown the ingredients as in the other recipe."
Lafcadio Hearn's
La Cuisine Creole (1885) includes eight different gumbo preparations, three of which are for oyster gumbo. One of them opens like this: "Fry a tablespoon of flour in a tablespoon of lard. Let it brown slowly so as not to scorch." Though not called out by name, that's a roux. It's added to two quarts of boiling oyster liquor along with onions, ham, and parsley. The oysters themselves are introduced at the very end along with a half cup of filé powder
How Roux Made Its Way Into the Gumbo Pot | Serious Eats
Prudhomme, as the article points out, made it trendy and the norm in NO restaurants.