No, the coat doesn't stop the seatbelt working. The coat causes the seatbelt to sit higher on the abdomen, over your tummy rather than over your pelvis.
It's designed to go over the pelvis as this is a strong area of the body. This is a good diagram showing where the seatbelt should fit on the body:
https://www.researchgate.net/figure/A-Proper-positioning-of-shoulder-and-lap-belts-with-respect-to-skeletal-anatomy-B_fig1_332067111
In a car crash the car stops moving but the occupants continue to move forwards until they are stopped by something - hopefully a seat belt and airbag. Effectively this means that the occupant is pulled back by the seatbelt. As this happens very fast, it would feel like the seatbelt is "cutting into" the person - you have probably felt a mild version of this in an emergency stop or minor shunt.
If it's over your pelvis then the pelvis is solid and it stops the seatbelt. You will be bruised and perhaps grazed but nothing worse (at least in that area). Your bum will also not move much on the seat, which is good (the less you move before stopping, the less serious the "third impact" - where your internal organs hit the inside of the chest cavity). But if it's higher in that gap between the pelvis and the ribs, there's nothing hard there to stop it, it will cut in with force to the abdomen. You don't want that, because there are a lot of vital organs in that cavity that can be injured by a violent intrusion like a seatbelt in a car accident.
Unfortunately it is the case that obese people are less well protected by a seatbelt than thinner people because the seatbelt will fit higher on someone with extra belly fat as well. This is also studied and documented. It's also the case that seatbelts protect men better than women on average, especially shorter women. This is because until recently all the crash test dummies were based on an average male adult body from something like 1960. There are recently female dummies and dummies based on obese males, which should hopefully begin to be used in order to provide better protection for anyone who does not fit into the "default male" body type.
https://www.bbc.co.uk/news/technology-62877930
We already know that seatbelts don't protect children as well as adults which is why children up to approx 10-12 years legally have to use booster seats. The booster seat solves the same problem. Without a booster, the seatbelt sits too high on a child's tummy in front of all the squishy delicate organs. With the booster, the belt runs correctly over their pelvis and can restrain them.
A thick cardigan does not usually cover the hips so doesn't have the same issue here. However if you have a very very young baby (like under 6 months) and the cardigan is very bulky compared to the size of their tiny shoulders then it can cause issues with the fit of the car seat harness (but not as much as a puffy coat or snowsuit would).