Many people in CLE didn't eat at Peking because it was owned by a non-Jew.
Then it was sold to a frum Jew from a respected CLE family and everyone started eating there. 2 years later he was busted for serving treif.
When a kosher restaurant is owned by a non-Jew, you would be relying 100% on the kashrus supervision, which might be more stringent. When owned by a Jew, the supervision might be more lax. I don't think it's possible to set a rule, one has to use שיקול הדעת. I don't eat out much, but I definitely care about both the Hashgocho and the owner. Though I must admit that when it comes to Milchigs I've mostly relied on the Hashgocho.
That being said, and tying to the discussion slightly upthread regarding Cholov Yisroel, I know a couple in Lakewood where the husband eats non-CY while the wife eats only CY. Now, I don't think something like that would fly where Kashering is required between non-CY and CY. For the same reason, if I see an establishment, even with a very reputable Hashgocho, that might have one or two non-CY or non-PY on the menu, I wouldn't eat in such a place.