I don't think Drebin is stereotypical and actually I think it's important to dissect the character for what it is and what it adds to the story rather then dismiss is as a racial issue and never get to the bottom of why he succeeds or fails depending on your opinion.
To put it simply I think Drebin is an overcooked character; there are too many attempts at personality. Contrary to most of the understated and subtle characters in the story he's not a believable character in the slightest. When a character isn't believable you stop caring about them, you refuse to invest any empathy in them.
He's your arm dealer, but he also works for the weapons manufactures, his name isn't real, and there are supposedly hundreds of them. Why such a dense tossed salad for a back story? This character would be fine if only half the above was used.
He's got a monkey, he drinks soda, he does magic tricks with a hanky. Coin tricks in front of Snake's face? C'mon! That is just tedious and douchey. Again there are too many personality traits going on here. Personally I would have kept the monkey because him trying to get a smoke is funny and the sound bite you get when you buy weapons is irreverant and whimsical. The soda drinking is dumb, the magic needs to be toned way down. Hanky waving should be for extreme situations only.
Finally is monologues are awful! Whether he's talking about the over the top conspiracies or the beyond melodramatic origins of a boss, these wind-sessions simply reinforces the clownish nature of the character.
Less is more. This wise adage should have been applied to Drebin.