You can always exhaustively run all Turing machines of a given size, at each step comparing the state & tape to all previous steps. If they ever reach the same state with the same tape as a previous step, they're in an infinite loop and never halt. If not, they eventually reach the halt state. BB(n) is computable. But BB is not computable, because there's no general algorithm requiring finite storage and time which computes the results of BB(n) for all n. You have to brute-force it for every positive integer n, which takes infinite time (and space) and is therefore not computable.

There is no requirement to have a recurring state when not halting. Indeed, the most trivial machine "write 1 and move right", will never have the same tape state.

Note that this casually solves the halting problem, usually a hint that something is wrong.