With ICBC they have to send everything to main Shanghai HQ to issue draft.
With Bank of China there is usually one branch in each district which can issue the draft on the spot. I’ve never been required to have a BOC account, but each area might have slight variations.
Also, I know that in Beijing they will not issue a draft on weekend. I don’t think it’s system issue more than a signature issue. Only certain people can sign the draft if amounts are over $1,000. Maybe those people don’t work on weekends.