This is very true.
It could be interesting to see what effect Bernadotte remaining in France has though. Sure, he and Napoléon didn't get on, with Nap dismissing him as a "second-rate general with too high an opinion of himself" (ironically, probably what Bernadotte likewise thought of Napoléon), and he was sacked mid-Wagram. Napoléon also said of him "Bernadotte hesitates at nothing [in battle]".
But when someone at St. Helena accused Bernadotte of treason, Napoléon responded "When he went to Sweden, he became a Swede. I can accuse him of ingratitude but not treason." (this was in line with Napoléon's own advice to his siblings when giving them thrones "conform to the customs of the country. Do not say "we do this better in Paris".")
Bernadotte joined the Coalition at the end of the Napoleonic Wars because it was in Swedish interests to so (Prussia, Russia and Britain would all have gladly taken something from them had they remained neutral or sided with the French and lost) - and Napoléon understood this, since when Napoléon released Bernadotte from French service, he asked Bernadotte to swear to never take up arms against France. Bernadotte told him that he could not swear something if it was against the interests of the country he was now chosen to be responsible for. Napoléon didn't push the point (unlike with Louis in Holland, where Napoléon had given him the crown, Bernadotte was chosen by the Swedes).