You're thinking like a present-day American and obviously are unfamiliar with both the Germans' modus operandi and the realities of life in an occupied country during wartime.
Wherever Germans and their local collaborators arrested Jews during 1939-1945, the doors of their houses and apartments were sealed shut. This was to keep anyone from looting the homes before the Germans themselves were ready to carry away the contents. The Germans stole everything they possibly could from the Jews they were enslaving and killing, right down to their hair and the gold fillings in their teeth. Monuments Men showed how the Germans stole not only artwork, furniture and jewelry, but even clothes that the owners were forced to leave behind.
So there's no "of course the neighbors would have checked" about it. Neighbors would have had no chance to enter the Starzynskis' apartment, nor any reason to. Had they tried, they too would have been arrested, or worse. There's also the ugly truth that many times, the Jews' neighbors were anti-Semites who preferred to not get involved.
Anyway, it wouldn't even have occurred to the neighbors to wonder if anyone was left behind. The roundup was carefully coordinated to take the Jews by surprise. Arrests began at 4:00 a.m. on July 16, 1942 when, like most Parisians, the Jews were asleep and unable to resist or escape. Their non-Jewish neighbors probably slept through the arrests and, when they woke up to learn about them, assumed that both Sarah and Michel were with their parents.
But for the sake of argument, let's suppose there were neighbors who did know the closet existed and Michel was locked inside it, and against all odds, were able to break into the sealed apartment and rescue him. Then what? Successfully hiding from the Germans, or helping someone else hide, required planning and could not be undertaken on the spur of the moment. Anne Frank's family began stockpiling food, clothing, and furniture in their attic hideout months before they moved in, as well as securing promises of help from the non-Jewish friends who brought them food and news while they were hiding.
Paris wasn't liberated until August 1944. The Starzynskis' neighbors couldn't hide a small boy like Michel for two years in a building with an anti-Semitic concierge, and they couldn't take him outdoors for fear of being stopped by police and asked to show their identity papers. Michel had no papers except those that identified him as Jewish. To get him false papers, his neighbors would have to pay bribes or go to the black market, either of which would cost them dearly.
Even then, the most skillfully forged papers would not protect Michel against the most cursory physical examination. Evidence of his circumcision would be damning because in mid-20th-century Europe, it was extremely rare for non-Jewish males to be circumcised.
Next, having taken him in, how were they going to feed him? The Dufaures were able to feed Sarah because they owned a farm and raised their own food. But for Parisians and other city dwellers, food was rationed, and to obtain ration cards, they needed to show ID. Their only alternative was the black market. Not only would Michel's neighbors have spent a fortune buying black market food, they also would have risked arrest every time they made contact with a black marketeer.
Finally, had Paris been bombed, they couldn't even go to a bomb shelter because police monitored shelters, looking for people with forged papers or no papers. If they went to the shelter closest to their home, they would certainly be seen by people who knew them, and knew they didn't have a child Michel's age. For a reward of money or extra ration cards, those people might turn in Michel and his neighbors to the police. If they anticipated being exposed this way and went to a shelter farther from home, the police would demand to know why. After all, with bombs falling, who wouldn't go to the nearest shelter?
No one would have courted so much danger and spent so much money for a child not their own.
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