My question is, what legal rights do I have when presenting my cashier’s check to US Bank for cashing?

A cashier's check is not a bearer financial instrument. It is made out to a specific name. When one is made out to you, its not like the bank has put the money (or attached it to the check per se). It is attached to it in a proverbial sense. What the bank actually does, is that it debits your account (when making the cashier's check), and places this amount in an Escrow account, until the check is cashed. When the specific check is cashed, the amount is released from the Escrow account to the account of the FI/Bank that is cashing it out.

What the cashier's check guarantees is that it cannot be bounced (or as the Americans put it - be returned for Non-Sufficient Funds).

So the cashing bank has every right to deposit and cash it the next day for you. They do this because there are so many fraudsters out there, that the bank prefers to deposit the cheque, have it checked out (no pun intended) and then cashed out to you.

The nature of the instrument allows the bank to refuse to pay it on demand, as it is technical a check that cannot be bounced, but a check nonetheless.

The answer - we don't have enough cash is a polite way of saying "We don't trust you / We don't know you and we will deposit it and and make sure this is genuine".