I delivered packages for UPS last Christmas season. And no, you do not always get discretion. Your board tells you whether you get discretion, or not.
Alcohol and wine deliveries require signatures. Deliveries to businesses require signatures (since they're supposedly open when we delivered to them, there is bound to be someone there to sign). Some bad neighborhoods require signatures. And yes, sometimes it's the shipper that requires a signature, for instance Best Buy always requires a signature (no matter how cheap the item is, or how good the neighborhood is, which is completely asinine).
In the case of the Amazon employee, I'll bet that he simply lived in a bad neighborhood, or a very large apartment building with no obvious safe place to leave the packages. In which case, this will get solved with the new pick up areas Amazon is setting up at liquor stores, supermarkets, and gas stations opened 24/7, or open very late.
As to Amazon, note that Amazon is only picking up the most lucrative part of the shipping business with its "Amazon Instant" and its "Amazon Fresh". It's focusing on instant deliveries, which even FedEx doesn't do (let alone UPS). This is the same way FedEx got started when UPS was already around, it focused on overnight deliveries since they could charge exponentially more for items that would get shipped overnight.
And by focusing on instant deliveries, Amazon can acquire impulse buyers that would normally go to brick and mortar stores to get their fix.