The verb you are looking for is “dit”. As per the Dictionnaire de l’Académie, 9

★II. Par ext. Faire connaître, exprimer, avec des nuances particulières.
☆1. Affirmer, prétendre

Here, its arguments are “les Français” and “portés sur la gaudriole et obnubilés par la “chose””, the latter being an adjectival syntagme, where the past participles porté and obnubilé are used as transitive adjectives. This syntagme could be replaced by any simple adjective

On dit les Français cannibales, mais…

The fact that this syntagme is long and contains transitive entities changes nothing as it is still only an adjectival syntagme, and doesn’t need any verb. Here is a quite informal (because I couldn’t be bothered to make it X’) and non-expanded tree of the clause

The predicate is “X are said to be Y”, where Y is “portés sur la gaudriole et obnubilés par la “chose”” and everything is fine, yay!

A quick primer:

As you note yourself portés sur la chose and obnubilés par la “chose” are regular adjectival phrases that are connected to Les Français. However, they do not use the common épithète or attribut du sujet constructions.

What we have here is termed an attribut du complément (by opposition to attribut du sujet). It is analogous to saying J’ai baptisé mon chien Fido., where Fido is an attribut du complément chien.

What the sentence means is something like

The French are said to be obsessed with […]

Which is quite different from your proposed Les Français sont portés sur la gaudriole (“The French are obsessed with […]”).

This sentence is somewhat hard to translate in English, but only because of the awkwardness of fitting a generic/impersonal pronoun as the subject in English (and the fact this particular construction sounds more natural with nouns than adjectives), but consider:

They called me a pervert.