Whether or not the 3DS will turn on is not strictly limited to the battery. It's a legit question. The only way the customer can test the battery, if you can't turn on the device, is with a charger.

And that's the point in time when you have to purchase a charger, as probably clearly stated in the user's manual. I'm pretty sure there's a troubleshooting page that says "try to load the battery" or something like that. In no way is it a case of warranty.

But on the other hand, I could see some people suing, because not everything needed to play is enclosed in the box, but that's a whole different problem.

I couldn't find any pics or gifs making fun of nintendo's tech support

I can confirm the boxes say things along the line of
'WARNING This console does not come with a charging adapter, you will need to buy one separately'.

As such it wont be part of warranty as it is clearly labelled about the need to buy the adapter.

...sad but true.