This really depends on the school district. We've lived in several different states and it's always been different in each place.
If your district does not regularly do one-on-one interviews with each new K student, you may certainly ask the school if you could set one up.
Generally, readiness tests consist of basic things like rote counting, counting objects, letter identification, if a child can balance on each foot and skip, if they can name objects from pictures and describe what that object does or is used for, naming basic shapes, writing their name, etc. They always seem to ask kids to draw a picture of a person and to copy drawing shapes like an X or T. Often they ask about tasks like zipping, buttoning, putting on a coat, and tying shoes (not always a requirement). Often there will be an eye test. Sometimes they will listen to your child speak and take note of any speech delays that they could help with.
Anyway, I'm pretty sure no public school can tell a student who meets age requirements that they can't sign up for K, they can just make recommendations yay or nay. IME, even if your child is light-years ahead of "normal" no one will really say anything about it. In fact, around here, if you happen to have a boy with a spring or summer birthday, regardless of K readiness, they will suggest you redshirt them! (A whole other topic!
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Of course, if you're talking about a K interview to get into a more exclusive school, then that's a whole other thing . . .