The frum people eat there on Shabbos?
Seriously, why can’t an establishment owned by a non-Jew be open on Shabbos? He has no obligation to keep Shabbos. If the kashrus certifying agency is able to make sure that all of their standards are kept (which may indeed be more difficult, but not impossible), then it’s doable.
Personally, I’d prefer to eat in an establishment which both has an owner who is shomer Shabbos and a yarei shomayim, and has a strong hechsher as well.