Can I use self-employed to file for an S Corp where I'm the only employee? Also - why can you not actually contact Turbo Tax to ask which product to buy?

Do you need to file the separate S corp return?  Then you need to use Turbo Tax Business.  TT Business is for 1120 Corporation, 1120S Corp, 1065 Partnership or 1041 Estate/Trust returns and will not do your personal 1040 return.   It  is a separate program from the Desktop Home & Business program or the Online Self Employed version.

Turbo Tax Business  is not available to do online or on a Mac.  You can buy the Window's (Win 7 or newer) version here.  And you can have both TT Business and TT Home & Business (or any personal version) installed on  your computer at the same time.
https://turbotax.intuit.com/small-business-taxes

You will get a Schedule K-1 from the S corp return to enter into your personal return.  Yes you can enter the 1120 S corp K-1 into your personal return.  

So why does the TurboTax Self-Employed version come with the 1120S form? Can't I use that?

You can enter a 1120 S corp K-1 into your personal return. But you can not prepare the 1120 S corp Business return.