It's extremely busy (as is the rest of NYC at present. I've never seen people 40 deep at a Macy's register! And last night there were 300 people lined up at the register at Target.) and the front desk is understaffed, but I wouldn't say unorganised. But definitely slow.
I've had packages delivered hundreds of times across the US. No hotel has ever refused. I do send them a message a few days before I arrive letting them know when it will arrive.
Update: Even busier than I thought. 25 mins to get a lift out of the building this morning!