“Chink in the armor” is certainly separate from the racial slur chink and is not racist, but it would be stupid (or perhaps just willfully disingenuous) to use it around or in reference to an Asian person.
I’ve never heard any of these “like a Turk” phrases, but just Googled and I see that at one point Amanda in the Glass Menegerie says “work like a Turk.” Must be obsolete?