It's a neat logical idea in theory, if you temporarily ignore any privacy concerns that some people may have. In practice though I can imagine quite a few problems unless other dramatic changes were also made to the entire postal system.
Firstly, the current address system is an international system that's generally used everywhere. Changing it in some places but not others would cause problems, although you could probably kludge the name as part of the address field. Also, simply in the US and other developed countries, keep in mind just how much of the existing infrastructure relies on people having addresses. Someone's going to have to go back and figure out what all those old COBOL programs do, for one thing.
Secondly, as you're proposing a lookup database for hundreds of millions of people, I presume you're envisioning computers and machines to be involved somehow. Obviously trying to expect people to look up that sort of information quickly is very impractical.
From the few month's experience that I had with a holiday job in a mail sorting centre (not in the US) several years ago, a large amount of mail is still hand-sorted for a variety of reasons.
We had OCR machines to do a lot of the work. Sometimes they misunderstood all of the ambiguous suttleties that people use in writing addresses, and that wouldn't be an issue if a name identifier were used instead. There were several other main problems that meant hand-sorting was still required though, including:
In the case of hand-written addresses, people's handwriting is often simply awful. I'm sure that if people realised that their letters would get delivered more quickly and reliably if they'd just print clearly on the envelope instead of scribbling, then most people would print clearly. As it is though, most people don't, and the machines dump a lot of letters into the reject bin for hand sorting afterwards.
Many letters and packages simply don't fit into the machines. These machines get through tens or hundreds of thousands of letters in an hour, but there's also a large variety of shapes and sizes of things that people send. People put all sorts of weird and crazy bulky things inside envelopes besides paper, and the OCR machines can't take all of them or they just get jammed. Those that won't fit also go to the hand sorting areas.
At some point during the process, your real address is probably going to have to be stamped onto the letter. (The postie needs to know where to finally deliver it.) The OCR machines could presumably do this, but it could be more problematic with hand sorting. Even if they have a hand-held device that they somehow enter the number into for an attachable label to be printed, it's going to slow down the hand-sorting process hugely. Having the address already written on the envelope by the sender at least allows an experienced hand-sorter to glance at and throw it in the right box very quickly.
I'm just speaking from a few months temporary work in a non-US mail sorting centre. I'm not currently a postal worker though, and I think it'd be great in theory for something like this to exist. On the other hand, it seems like there would be a lot of practical problems to overcome before implementing this -- it'd be a bigger change to the infrastructure than what initially appears. If anyone in a better position to do so would like to claim otherwise then please do so.