There was an attempt in the US, with the whole concept of the melting pot. In the interest of maintaining the other cultures though, this has been phased out in favour of the "salad bowl".

Imo, the catch is that by prioritizing ethnic cultures, you then allow them to supersede the national identity. Ethnocentricism is the default thanks to our monkey brains and in the absence of reinforcement, or the presence of discouragement, it'll probably win over other identities.