The French colonial empire, especially during the late 19th century and early 20th century, pursued a rather aggressive "mission civilisatrice" to spread French culture and customs throughout her empire. The French imposed their laws, their way of life and their language onto the conquered peoples, in the hopes that they too can become somewhat "French" like them as well. This heavily contrasted the kind of colonialism that Britain pursued, which was more economical and pragmatic in nature, and preferred to rule over her territories by keeping local rulers and traditions in tact, for the most part.
In the case of French colonisation we can see its effects linger in francophone Africa, where French is still used as an official language in many countries and is spoken alongside the local languages. Even in the Mahgreb, though Arabic remains the sole official language, knowledge of French is prevalent and learning French is common at school.
So why is it then that the French language is no longer commonly spoken in the regions which were previously French Indochina? I would have thought that France's aggressive mission civilisatrice would have had a large impact at least on the language.
I myself am Indonesian, and I can speak for why virtually nobody now in Indonesia speaks Dutch: the Dutch themselves never bothered to impose their language and preferred to speak in Malay (the lingua franca of the region) with native Indonesians. Only the Indonesian elite learned and spoke Dutch.
This is a really interesting question! You are quite right that after the 7 Years War, France's imperial "style", so to speak, relied heavily on assimilating cultures they colonized with French culture. This was a policy begun in Africa. The scramble for Africa had left France with vast swathes of low-population, culturally non-homogenous, largely unexplored land. To organize this territory, the French drew more or less arbitrary lines through the desert and jungle without much regard for ethnic, linguistic, or other local cultural markers. This policy in turn left the colonial governments to oversee a demographic mess, which was quite unstable as you might imagine.
Now the British had faced similar situation, and as you mention, their policy was usually to choose one group, give them lots of privileges and political power, and use them as local intermediaries to keep the peace in the region while British commerce and industry exploited local resources and markets. We see this with the Shia in Iraq, the Barmar in Burma, and a multitude of other examples in India.
The French response was quite different. They chose to replace local customs with French customs, and offered an array of incentives and penalties to this end. Children who learned French might gain access to University education in France and positions in the colonial government upon graduation, while those who resisted French assimilation might face attack, serfdom, or forced servitude (slavery in all but name) in French households. While brutal, these methods did serve to cement the French language as the language of the elite and the intelligentsia all across Africa, from Senegal to the CAR, because the arbitrary lines that the French drew remained more or less in place after decolonization, and this unifying language was necessary for the continued administration of culturally diverse regions.
Now, on to Southeast Asia. We must remember that the French arrived in Indochina pretty late in the game, establishing control over Vietnam and Cambodia in 1885 and Laos in 1893. They proceeded to implement a similar scheme as to that which was implemented in Africa, and to some degree this was effective, but French control of the region was shortlived. Whereas Africa was colonized from 1830-1960, Indochina was only under French control from 1885-1940, about half as long. Furthermore, decolonization occured along well-established, non-arbitrary lines, i.e. what had been Vietnam pre-1885 remained Vietnam post-1940, what had been Luang-Prabang pre-1893 became Laos post-1940, etc. To your question, everyone who lived with the borders of Vietnam had been speaking Vietnamese for a long time before the French arrived, most of them continued to speak Vietnamese while the French were there, and once the French were out there was no reason for the political elite to continue speaking French. The same situation applies to Cambodia and Laos as well.
If you are interested in the subject of French Colonialism in Southeast Asia, I would highly recommend looking for the following titles: Vietnamese Tradition on Trial and Vietnamese Anti-Colonialism: 1885-1925 by David Marr, War Comes to Long An by Jeffery Race, and Imperialism in Southeast Asia: "A Fleeting, Passing Phase" by Nicholas Tarling.
tl;dr: Most people spoke Vietnamese, Khmer, and Lao through the colonial phase anyway, so one the French were gone there was no reason to keep speaking French.