Does "at any given day of a week" make sense here?

"On any given day of the week," but otherwise, yes that conveys what you want. "On any given day," is terser, and perhaps clearer if you really just mean a random day. "On any given day of the week," emphasizes that it doesn't matter whether it's Tuesday or Sunday or whatever, but has the same broad meaning.

I'd say, "...on any given day of any given week", if you want to mean a random day of a random week. "At any given day" doesn't seem correct here, you borrow things on a day not at a day. (Native speaker, AmE, New England)

No, it should be "on any given day"

Give us the context, and we can rewrite it. How many books can you borrow? Is their a daily limit, or... well... what are you trying to say?

Not really. You should just say, "on any given day." I don't think I've ever heard the phrase "on any given day of the week" (and it would never be "on any given day of **a** week").

If you mean a random day on a ramdom week you can say "any given day of any given week" which I think makes more sense and conveys your meaning better than most of the comment sugestions here (if im correctly understanding what you want to say)

A random day already implies a random week. I think the following work: - "on any given day" - "in a typical day" - "on an average day" - "per day"