alt Hacker NewsOK, I'll bite. In this particular case I personally would have said "a restaurant of your national cuisine" or even omit the adjective altogether since it's somewhat obvious from the preceding conversation what you mean.
My main qualm with this usage -- aside from being dated -- is that all cuisines are ethnic, but singling out the hypothetical migrant's as such implies that it is particularly ethnic. So in this sense it is a bit othering.
To be clear, I don't write this to call you out for your usage or accuse you -- I'm just answering your good-faith question.
RE your point, I don't agree that it's so obvious -- cooking and running a restaurant are pretty hard regardless of where you are from!
Here ethnic cuisine more or less means regional cuisine, where the region is outside of the speaker's own cultural identity. It goes beyond national, which is very broad. Therefore I think it's a fairly useful descriptor.
For example Oaxacan and TexMex are ethnically Mexican, but seen as regional within their borders by the respective nations. There's deeper indigenous roots there too. Food traditions of the Jewish diaspora can't be said to be Israeli cuisine. Things that start as regionally unique become seen as ethnic over time: I dunno like rocky mountain oysters, fish and chips, buffalo wings, as they become more intertwined with the culture of those people and that culture is exported.
Ethnicity was coined to avoid the racial connotations but in the US, ethnic studies brought it right back. If one wishes to avoid the word here maybe cultural is probably the best substitute but still a bit too narrow. You'll also see Global/International/Continental/Intercontinental as well, much like in music.. but I think that misses the point too.