I managed to connect the ATV to ethernet. I also connected to a different router with wi-fi and ethernet.
The results? VERY surprising!
Turns out, I CAN get the ATV to stay ping-able when in sleep mode. This is true on both routers, and it is true on wi-fi and ethernet. However, the router must be disconnected from the Internet, and then I must restart the Apple TV. When the ATV comes up, there is no normal "home screen" full of icons at this point, only "Computers" and "Settings" icons. I can then go into settings and put the ATV to sleep. Then it stays pingable while in "sleep." It is asleep right now, and I have been pinging it for hours with Internet disconnected. (ATV is connected to router via wi-fi, Internet connection is unplugged from router. ATV can see local network only. No Internet access.)
I also tried restarting the ATV while connected to the internet, then I disconnected the Internet from my router. Then I put the ATV to sleep, and then ping STOPS responding within a few seconds. Then when I reconnect the internet (while ATV is in "sleep"), it notices that it is connected to the internet, and the home screen apps load up in the background. Then it STOPS responding to ping.
To recap - when ATV is booted normally and connects to the internet and fully loads, then putting it to sleep makes it stop responding to ping. But if I restart the ATV with no internet, it does not fully load, and comes up with a more limited selection on the home screen. In this state, if I put the ATV to sleep, it remains pingable for hours - maybe even longer. It seems to have nothing to do with the router, and nothing to do with wi-fi vs. ethernet. Then if I plug my Internet back into the router, the ATV is then able to load all the way (while sleeping - blue LED is off and still pingable), and then it STOPS responding to ping. The only way to keep it responding to ping while in sleep mode is to keep the internet disconnected from the router.
This could explain why there is such a mixed bag of user reports on this issue. It's hard to nail down what is happening. I just did a full restore to factory defaults. When it was done restoring, without changing anything, I put the ATV to sleep, and then within 10-15 seconds, it stopped responding to ping.
The only thing I can think of to try next is to order a new ATV and see what that does. My feeling is that this problem has to be specific to the hardware of certain ATV units, but I guess I really don't know. Swapping out the unit should shed some more light.