...Can someone please explain why ZIP codes are even a thing people need to bother about? Like, there's an address on the letter/parcel — that should be enough for the USPS to deliver it, right? If they use some internal classification/address grouping for better routing — fine, whatever, they can use it but surely they themselves can look it up in their internal DB quicker and more reliably than I can on some random web-site of dubious progenity?

I assume you're asking this question in good faith. A mailing address alone is often ambiguous, eg tons of duplicate street names, people write the wrong city, and typos and missing unit numbers are common. The ZIP code doubles as a sanity check against the rest of the address. USPS will usually figure it out without one, but it takes extra time and effort. Also, a person's zip code also doubles as card verification and sales tax. Maybe the opposite question should be asked, why do we need addresses? Your residence could be defined expanding on its zip code.

It made a lot of sense in the pre-internet days. Getting a human to sort mail without looking up a big list was pretty useful.