You can use an alternative layout by setting the layout attribute it in the JSON template, instead of using {% layout none %}. It gives you the access to use section on the page. The layout attribute is mentioned here: https://shopify.dev/themes/architecture/templates/json-templates#schema
I am now trying to get the template editor to work properly with the page. For some reason it edits the new page template, but saves to the “default page”.
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I also found the solution to the “default page” problem I mentioned above: If you put a link to the page somewhere on your main page and navigate to it through the theme editor tool, the tool will correctly interpret the page template you are using.
The problem I mentioned with “default page” happened because I went to a page that uses the default page, then used the ‘change’ link (in the upper left side of the screen) and selected the page that uses an alternative layout. In this case the theme editor tool does not interpret which template to edit correctly.