The reason the customer should check into it first is in the event the package is going to be redelivered or the package was delivered but the customer either didn't actually check for it or it's in an area the customer didn't look.
I have a dear friend who works PT for a major company in CS. She was shocked at how many customers call in demanding to know where their package is because it's supposed to be there and then informs her (when she tells them it was delivered) that they hand't checked yet or they are at work and will check when they get home. Often a note is left by the PO because of an issue with delivery. That isn't her company's problem; that is the customers and they need to make arrangements for the next delivery (because they know THEIR schedule and what works for THEM).
With UPS using USPS Surepost, the information relayed to CS is often behind so they know nothing more than what their screen shows.
Last, if you are provided with the tracking #, you can track it. You see the exact same thing CS sees, believe it or not. Once you've tracked the package and there is an issue, you relay it to CS and then they take it from there, she said.
Last, fraud. People claim they didn't receive something but have. By making the customer take responsibility for locating the package first, it weeds out those who know they can't make a false claim. It saves the company so much in fake losses.
You may or may not agree but this is her perspective from working CS. I had no idea until she told me. Now you do too.
*Four Seasons once again*