Priority carries a per-screen flag that decides whether the API licence may touch that screen at all.
A screen without the flag is not slow over REST, and it does not return a permission error that points at the cause. It is simply not there. Because the flag is set per installation, the answer at one customer is not the answer at the next: an integration that shipped last year can be impossible this year at a different site, with identical code and nothing visible from outside to explain it.
Establishing the real list is a read-only check that takes minutes. Running it before the design rather than after the estimate is the cheapest hour on the project.