I am not sure how conceivably anything is "wasted".
You need to pay 12 payments of the 20 payments to turn the phone in. So... pay those twelve payments day one. Now you can get a new phone when the time comes. All you have done is decided for yourself not to make the payments monthly, but rather pay up front. No money is thrown away, you just paid today rather than tomorrow.
If your phone gets lost/stolen, or irreparably damaged, that is a chunk of money you don't have to deal with at that time, because it's already been paid. If you have the money to put down on day one, you are literally not losing out on anything that a person using the monthly installments is getting (unless you count giving up 0% credit as missing out).
OK, since no one has definitively determined if AT&T will allow you to put a down payment on the purchase of a phone, my reply will just be a thought exercise.
The OP talked about putting a down payment on a phone in order to lower his monthly payments. Let's say he's buying an iPhone 5s 16GB and wants to spread his payments out over 20 months. If he goes with $0 down, his payments will be $32.50 over 20 months for a total of $650. At the 12 month mark, he's eligible for an upgrade, provided he hands in his phone. He's paid $390 already and AT&T will allow him to forgo the remaining $260.
Now, let's say the OP wanted to put $100 down. His monthly payments will now be $27.50 over 20 months. At the 12 month mark he's paid $330, plus the $100 down payment. AT&T let's him off the hook for the remaining $220.
In this scenario, he's out $40. The larger the down payment, the more he wastes. Keep in mind, this is just a thought exercise. Maybe AT&T would allow you to take your $100 down payment back or something like that.
I haven't read the fine print of the T&C's, but the promotional material states that you can upgrade when you've made 12 (or 18) monthly payments. Your premise of making 12 payments on day one may not apply here. It's really semantics. If you put $390 as a down payment, you've technically not made any monthly payments. If you opt for $0 down, and paid $390 on the first bill, then you may have a point.
Either way, putting money down in the traditional sense of a down payment is foolish if you're able to get 0% financing. Even if the interest rates are low, it's still money your throwing away. Not to mention you lose cash flow flexibility in case you needed that money for something else, but now you've given it to AT&T unnecessarily.