TL;DR. See the accepted solution below, but in short: you cant do it!
OP frustration: it's infuriating that I could easily get to very clear documentation suggesting that this was still possible, while approaching this from another direction leads to equally clear documentation saying that it's not.
I'm trying to use Google Scripts to get (simple) access to the Sheets API. Tutorials on how to use Scripts from within a Google Sheet work fine for me. But I'd like to have the script run from a Google Site (so I can construct some HTML to display a result to the user). And I'm stuck really early in the process.
In my reading of the google docs, this link: 
CodePudding user response：
All I can say is that 
Even the actual API is limited to the classic Google Sites:
Sites API:
If what you want is to have a script that shows data from sheets to your site, what you could do is create a webapp and then embed it to your site. See references for more information.