What is an orthodox shul? You are defining them as orthodox first, and then saying that anything they do must be orthodox!
You should first look at what they do, and if it is orthodox, then define them as orthodox.
A shul that has no mechitza, is not orthodox. Even if it is Beth HaMedrosh Hagodol-Beth Joseph in Denver and is affiliated with the Orthodox Union. (They recently hired a YCT grad, if that fascinates you).
A shul that has women davening for the amud, or getting aliyos, or layning–is not orthodox. They may be very nice people–and many of my friends go to such shuls and are very nice and sincere people. But they are not orthodox.
And a shul that hires one of these guys (or gals) is not orthodox. You can ask your own shaila about if you are allowed to walk inside or daven there; but that won’t make them orthodox.