There is very good reasons why De Gaulle was always a bit doubtful about American military protection and why post-war France put a strong emphasis on military sovereignty.
That has nothing to do with any French stubbornness or a so called French anti-American feeling.
The main reason is that De Gaulle experienced the fact American leadership can be untrustworthy first hand.
When he was the leader of the exiled French force during the 40s, Churchill supported him.
Meaningwhile Roosevelt refused to give him any support and actively acted to make him replaced by a puppet, General Giraud. Mainly because it was better aligned with American interests to setup a puppet state in France on the longer term.
The situation changed only later when it became pretty obvious that Giraud was antisemite, an openly nazi collaborationist and a pretty poor politician.
Only then, America started to support De Gaulle officially. Initially only indirectly through the relation between De Gaulle and Eisenhower.