Who was to hold it?

Nazi ideology always meant that a struggle in the East against the USSR would ensue, and Great Britain did not yield following their disaster on the continent. This created a problem; the coast of France would now have to be held in more than token force, in addition to the necessary military occupation. By allowing what was essentially a puppet French Government to form, Nazi Germany cut down on quite a bit of terrain they would be directly responsible for, and caused confusion and dissent viz. the Free French situation. The Vichy French government was born out of pro-Facist and anti-Bolshevik sentiments in the French republic, and it made no sense not to grasp onto these sympathetic politicians for military and political purposes. The illusion of legitimacy created by Petain's government greatly delayed the need to send significant German occupation forces to the south of France for anti-partisan and subjugation duties. The drain on manpower was far less severe as a result, and Nazi Germany required every formations they could on the Ost front.

For all of Germany's spectacle and might in the early years of the war, and despite its impressive Army size, even Germany had to cut corners when it came to occupation. Germany had big ambitions in the East, particularly in the USSR, and troops were perpetually needed everywhere, given that Germany needed every soldier it needed to try and survive its numerical disadvantage against the Soviets, and then then Western Allies after 1944.

The difference in France than in Eastern Europe, is that France had a decent amount of pro-fascist politicians and sympathizers. Whereas in countries like Yugoslavia, Poland, the Baltics, resistance to German occupation was much more acute and severe, strong occupations had to be enforced, while Germany could afford to let France have semi-independence due to the sense of legitimacy these sympathizers would create. It wouldn't be until 1942 that the Germans would officially occupy Vichy France as well, following the Allied invasion of French North-West Africa.

For a little bit of perspective, throughout 1943 and 1944, when Germany was getting overrun on both sides, it kept sizable occupation forces trapped in Estonia and Norway, both of whom were reasonably well trained and could have helped Germany on any of its three fronts. However, since Estonia and Norway had to be kept under control, these troops were kept there despite being needed elsewhere. This is the cost of having to enforce an occupation over a country, your pool of manpower goes down because you need soldiers to enforce an occupation. France is a comparatively pretty large country, is significantly more densely populated than most other European countries, and has a much more militaristically oritentated mindset that was bound to explode due to the nature of French-German animosity at the time, therefore, the chance of pacifiying France through its pool of pro-fascist sympathizers, and minimizing the amount of troops Germany would have to commit to it, was the best solution, even though these factors would help force Germany to finally force an occupation in 1942.

Basically, in short, like BritainOpPlsNerf said, Germany needed every soldier it could use for the Eastern Front primarily, so corners were cut and the use of occupation forces was minimzied as much as possible.