I can confirm that the Project Page Generator is not available anymore. No matter if you create a User or organization site and have the website in the repository's master branch or a Project site with the website in a gh-pages branch. It works differently nowadays and it also has a different name.
The so called Project Page Generator is as you might have noticed no longer promoted. If you have a look at the GitHub Pages promoting page from the past using an internet archive. you will see that this feature was promoted in earlier days like:
Project Page Generator
If you don’t want to go through the steps above to generate your
branch, or you simply would like a generic page, you can use our page
generator to create your gh-pages branch for you and fill it with a
I assume you already activated GitHub Pages in the repository settings.
For standard static HTML following the guide on GitHub Pages is good enough. You just push a HTML file into the repository and it will be online just after a few seconds.
For markdown however the easiest way to get started is to choose a GitHub Pages theme in the repository settings using the Theme chooser. Once selected use the index.md of your chosen theme as a starting point. You will find it at https://github.com/pages-themes/. With the syntax described you can directly create webpages within GitHub.
As far as I know for local testing the default layout file within the layout folder has to be copied from the theme's repository too. But this is definitely not required if you work on GitHub only. Unfortunately the themes seem not to bring any layouts for blogs. You could create your own of course.
Nevertheless I recommend the official documentation of Jekyll to GitHub Pages. There you find a very helpful link which is a very good step-by-step guide from Jonathan McGlone. With this code put inside your website repository you will have a basic structure for both static sites and articles using a basic adjustable theme.
Another tip: A good way to create and edit your posts from within a web interface is prose.io