No, you don't pay a premium for next day delivery. You pay a premium for unlimited delivery for free for any number of orders. Other people pay per order for next day delivery.
Prime is you buying next day delivery in bulk, nothing more or less, no different to buying next day delivery one at a time. You are now also paying for other things with prime like music/video streaming.
Guess what, when order volume goes up you get delays, people that don't understand that and instead say it's disgusting should be banned from shopping online. Go to a store during a busy period and compare how long it takes to get served compared to non busy period.
Not entirely correct, whilst the prime membership advertises as "unlimited one-day delivery". When you put through an order, they give you a guaranteed delivery date (next day) something they've failed on multiple times this week. This is what i pay a premium for. The music/video streaming service is just to sweeten the deal for people who might also be interested in that.