There is a lot of confusion regarding the French Gendarmerie, fueled by descriptions like “military police” or “military force with law enforcement duty” which are technically correct but do not fully reflect the nature and tactics of the force. Its members are indeed military officers, which has some legal consequences (trade unions are strictly forbidden, ranks are different) but it is otherwise a regular law enforcement agency, not an elite force or second level of response when the police cannot deal with a situation.
In general, the French Gendarmerie is policing rural areas and small towns when the national police (Police nationale) is in charge of big cities (and above all Paris, which the French government has long feared as the source of many revolts and disturbances). In its area of responsibility, the Gendarmerie covers the full scale of law enforcement activities, from policing traffic, securing road accident sites (a major role as they cover most of the long-distance road network) and response to emergency calls and disturbances (police secours) to criminal investigations and riot control.
French gendarmes are not issued and do not routinely carry military weapons (e.g. HK 416) but simply a handgun (Sig-Sauer SP 2022, procured through a common call-for-bids covering both police and Gendarmerie). Unlike the rest of the military, ranking members of the Gendarmerie have power-of-arrest (officier de police judiciaire) and are trained in criminal procedure. They will act at the behest of the justice system when appropriate (executing warrants, etc.) In this like in so many other aspects they are closer to the police than to the military.
Operationally, both the police and the Gendarmerie have SWAT teams (the GIGN and RAID are the most famous). It is extremely rare for the Gendarmerie SWAT team to be called in to intervene in the police area of responsibility and vice versa. Both the police and Gendarmerie have roughly the same number of riot control units (unité de force mobile: escadron de gendarmerie mobile in the Gendarmerie and compagnie républicaine de sécurité in the police), with similar tactics and organisation, deployed side-by-side depending on availability (especially in Paris). The Gendarmerie does have a few aging armored vehicles, seldom seen in metropolitan France but that's about the only difference when it comes to riot control.
The Gendarmerie has a few capabilities the police doesn't have but they are really marginal and not directly relevant to your question: mountain rescue, coast guard duties, security for air bases and nuclear sites, provost duties for the French military at home and abroad, deployment abroad to secure French military bases and diplomatic posts. The GIGN also has a long tradition of training and working with the special forces and has officers trained in military parachuting or assault at sea so that it is tasked with some counter-terrorism missions on the whole territory (i.e. even in the police area of responsibility). The police also has a few specific duties that are not shared with the Gendarmerie: investigation of terror attacks, border control and railway police, larger role in domestic intelligence.
All that is a long-winded way to say that the analogy in the question is not the right one. To the extent that there are debates on a “militarization” of law enforcement (and in France there are, for example regarding riot control tactics), it would not focus on Gendarmerie vs. police. If anything, the Gendarmerie has a more positive image in the public and a reputation of being less “heavy handed” than the police. With a few key exceptions (Rémi Fraisse and Adama Traoré), all cases of excessive violence and racism by law enforcement that have created controversy in France in the last decade involved the police, not the Gendarmerie.
Incidentally, while they do sometimes retain a different legal status and traditions, there are signs of a convergence between Gendarmerie and police in many European countries. In Austria or Belgium, the Gendarmerie was merged with the police. In France, it was transferred from the ministry of Defense to the ministry of the Interior.