Is the mashgiach reliable?
Who’s opens the doors to restaurant?
Who’s lighting the fires?
Who’s opening/pouring the [non-mevushal] wine (outside and INSIDE the kitchen)
Are “Gush” greens being used? Who’s making sure they’re being washed?
Orlah can be an issue for olive oil (if you’re already concerned about terumos & maaseros).
How on top of the audits is the mashgiach to make sure nothing else is getting in?
The milchig foods themselves might rely on kulas you might not rely on.
That’s the short list; hatzlacha 🙂