> The electrical outlets in your home have two power-carrying prongs.
Only one of the prongs carries power. The other is attached to the earth, and is always at zero volts.
> and then they will slowly switch places over the next 1/120 of a second so the right wire is at +170 volt
No. The power prong (the smaller one) will switch from -170 to +170. The other one (the neutral) is always at zero (measured relative to you).
> Turing three-phase voltage into two phase voltage is pretty easy. Just pick any two of the three wires and hook them into the home.
This is not what they do. If they did they would not be able to have a neutral, and you would also not have an option of 240v. Homes don't have two phases, they have split phase.
Instead they take a single phase, and attach it to a center tapped transformer, which I described here: https://news.ycombinator.com/item?id=16380133
> Just pick any two of the three wires and hook them into the home.
In actuality the phases have 240 volts between them, not 120. And from any phase to neutral is 208 volts (which some devices make use of). (Those are voltages to customers. Internally they use other voltages - in particular when they use a single phase, power a street with it - they care about the voltage to neutral, not the voltage relative to another phase.)