I always thought Apu was somewhat fleshed out for the stereotype characters: he's not just the Indian store owner. We learn that he's a bit of a Playboy and a ladies man before he marries manjula, he's vegan, he likes to sing, he's a volunteer firefighter, and generally well intergrated into the community. He has development beyond his race in the fact that he gets married, has kids, has an affair. He's not serve tile as the documentary states, he is polite but does have a wider range of emotions when pushed.
It's more than the likes of Groundskeeper Willie (and drunk Scotsman) Fat Tony (Italian American Mobster) Lindsey Naegle (cold, stern, caonniving business woman with commitment issues)
It all seems ridiculous and to be honest, of the stereotype Apu seems the most human and real