I've said it before, but the main obstacle to analyse French as polysynthetic is that it doesn't satisfy the "morphological visibility constraint". Not only all arguments are not marked on the verbs, but in fact, modern colloquial European French has even a tendency to drop clitics that would be considered obligatory in literary French: "je vais le lui donner" often just becomes "je vais lui donner", with the direct object entirely unmarked, simply implied by the verb (donner = to give). In some instance this can go further to "vais lui donner", with dropping of the subject clitic as well (though in this case the verb does have a separate, audible inflection for 1sg, but that would work with a less marked verb: "viens d'lui donner" = "(I) just gave (it) to him/her").
When you get into imperatives, even literary French is not shy of things like "Donne !" for "Donne-le-moi !" ("give it to me!").
I'll be happy to defer to you on that topic since (I believe) you have formal training in French linguistics, and I haven't.
I don't believe the analysis of spoken French as polysynthetic either - though I'm having a lot of fun deriving a polysynthetic language from French, but there are a few things I disagree with in your analysis.
The "je vais le lui donner"
> "je vais lui donner"
change is almost systematic, however I wonder if that wouldn't be a case of phonemic reduction.
(The schwa in le
ought to be dropped, leading to something like [llɥi]. It doesn't seem unreasonable that it would get further reduced to [lɥi].
The similar construction je vais te le donner
(substituting a second person to a third person) cannot, however be reduced to *je vais te donner
vais lui donner
et viens de lui donner
sound ungrammatical to me. I'll try and pay attention, maybe I'll hear it used.
I'll argue that Donne !
is more a literary French construction than a spoken one.
The one thing that would prevent an analysis of French as polysynthetic is that head-marking constructions such as (a) 'Le livre, je te l'ai donné'
happily coexist with boring ones such as (b) Je t'ai donné le livre
(I believe this is the point of the article Whimemsz links to, which I haven't read in full yet)
(In my conlang, I posited a series of changes that turn (b) into a single verb, with livre
as an incorporated nominal)