Asking for the promotion isn’t arrogant. Assuming they’ll notice without being asked is the risky part.

Managers are busy and generally do not keep track of who is ready for more. Working quietly and waiting to be recognised feels modest, and it mostly hands the decision to somebody with other things on their mind. Saying what you want is information rather than a demand, and it is the ordinary way these things happen. The risk sits with the silence rather than with the asking.

Work out exactly what you want, whether more hours, a different task or a key to the building. Say it once, plainly, with one sentence about why you are ready.