This isn't really a security question as much as a networking question. That said, if all a router could know was the next/previous router in the chain, how would the next host know who to send the packet to?
Think of network packets as you would postal mail. Imagine if the original sender put their packet into an envelope with the address of the nearest post office on it, instead of the address of the destination. How would the post office know to whom to eventually deliver your mail? Worse, imagine the letter is somehow delivered to your recipient. How do they know who to reply to? Each post office along the way would have to remember every piece of mail that went through it previously, to associate a reply back to the original sender.
Perhaps you might include nested envelopes, with each next envelope listing out each sorting facility along the way. How are you supposed to get the canonical list of hops between you and the eventual destination? What if one of those sorting facilities is overloaded, or worse, no longer operational — how can the postal system reroute your mail if it has the entire path through the system written down in every envelope? This would also be expensive and costly! Postal employees would spend a massive amount of time opening envelopes just to see to where each piece of mail needs to go. And again, what happens when the recipient wants to reply? How do they specify the routing of your mail if they don't know its eventual destination?
So if you can't just include the next hop, and it's completely impractical to include the entire list of hops, what's the answer?
Obviously, you put the destination address on the envelope along with your return address. Every hop in-between makes an independent decision about how to get your mail closer to the destination. If a new sorting facility is opened, an existing one closed, or if there's a capacity problem that can be resolved by going through a different route to the destination, the postal workers can figure out what needs to happen. If mail gets sent to the wrong facility by accident, it can still eventually arrive at its intended destination. Processing mail can be fast — there's no need to open envelopes — they can simply be passed through the system untouched. And when the recipient gets the correspondence, they can reply with a single piece of information: who it's going back to.
Unsurprisingly, this is how the Internet works. You include the destination address and the source address in every IP packet. All a router needs to do is quickly look up how best to get your packet closer to its intended recipient, and then they can statelessly forward that packet, completely unaltered, down the line to the next router.