Google Sheets API: How to "publish to web" for embeddable sheet?

After searching through the entire Sheets API, googling, and just general soul-searching, I had no choice but to include the Drive API and use it to do my bidding. Here's the solution I came up with. Hope this helps someone else out there!

Used this script from Google for the client-side JS library in the index.html file:

<body>
  ...
  <script type="text/javascript" src="https://apis.google.com/js/client.js"></script>
</body>

Then for the JS stuff:

// Cache the api's into variables.
var sheets = gapi.client.sheets;
var drive = gapi.client.drive;

// 1. CREATE NEW SPREADSHEET
sheets.spreadsheets.create({
  properties: {
    title: 'new-sheet'
  }
}).then(function(newSpreadSheet) {
  var id = newSpreadSheet.result.spreadsheetId;

  // 2. PUBLISH SPREADSHEAT VIA DRIVE API
  drive.revisions.update({
    fileId: id,
    revisionId: 1
  }, {
    published: true, // <-- This is where the magic happens!
    publishAuto: true
  }).then(function() {

    // 3. DISPLAY SPREADSHEET ON PAGE VIA IFRAME
    var iframe = [
      '<iframe ',
      'src="https://docs.google.com/spreadsheets/d/',
      id,
      '/pubhtml?widget=true&headers=false&embedded=true"></iframe>'
    ].join('');

    // We're using jQuery on the page, but you get the idea.
    $('#container').html($(iframe));
  });
});

As you have concluded, it is not possible through the Sheets API today and is only possible through the Drive API (using the PATCH https://www.googleapis.com/drive/v3/files/fileId/revisions/revisionId request, documented at https://developers.google.com/drive/v3/reference/revisions/update).