Seriously?
It was the weekend and I have better things to do with my life than sit on a forum.
You will not get a higher frame rate on console games playing at a lower resolution.
The games are rendered in a lower resolution then upscaled in order they can be viewed at 1080p (or whatever resolution you're at).
The kid even said that himself.
Therefore, if you set your console to 720p, it just means that the rendered frame isn't upscaled.
It doesn't increase your frame rate.
It doesn't make your console "run faster".
Furthermore, console games are usually locked to a frame rate, so even if the console COULD render at a higher framerate (from lowering the resolution), then it would still only display at the locked frame rate.
It wouldn't run faster.
How do you want me to back up what I said?
I can't find anything from a quick google that "proves" this, because no one has investigated it.
There's no need to investigate it.
Consoles usually run games around 30fps, and lowing the resolution will unlikely have any effect on this.
I looked a little harder and found THIS.
There's absolutely no need to be rude.
Have a nice day.
EDIT: Just to cover my ass, yes there are some exceptions where the console will increase frame rate when decreasing resolution, but they are few and far between