My rebbeim taught me that if you wouldn’t eat in someone’s house, don’t eat in their restaurant (or under their hashgocha).
I wouldn’t eat in a home where the women don’t dress according to halacha, or if the men disregard halachos of shmiras eynayim altogether. Why would you trust someone who doesn’t accept mitzvos as a Mashgiach? Why are they believed to be relied upon when they are invalid as witnesses? This includes, according to rav vosner, someone who has unfiltered internet.
There are plenty of restaurants with Mashgichim/owners who wear knit yarmulkes, who i have no problem eating in, but I wouldn’t eat in a place if i saw the Mashgiach watching a movie on his phone, regardless of how he’s dressed.