Hi Neal
I visited your staging site to try and see for myself what the problem is.
From your last comment, that disabling Layouts meant that you were again able to select the Layout settings that you could not while Toolset Layouts is active, it seems you are still conflating Genesis Layout options and Toolset Layouts, as this is entirely expected, and I encourage you to re-read the Toolset Layouts documentation (https://toolset.com/documentation/user-guides/layouts-getting-started-guide/) and to also read the Genesis framework documentation which should hopefully clarify the difference for you.
You should also think about what you are using Toolset Layouts for, because if it is not needed then there is no need for you to persist in trying to get it to work. If you can already achieve what you want with your theme (given that it includes extra layout options for with- and without-sidebars etc.) then you may be able to achieve the required design just using your theme and Views (using content templates, custom archives, and Views added to widgets) without the added complication of Toolset Layouts.
Sometimes people use Layouts because its there without having a real use case for it (which is to be able to customise how content is added to a page in ways which is not otherwise possible).
Having said all that, you are correct in that pages using Layouts are not rendering as expected.
You will find that I disabled all non-Toolset plugins to abstract from any potential conflicts. I then created a test page (Toolset Test page) and a test post (Toolset Test post) to which I assigned a new crude Layout (Layout for Toolset Test page) which outputs just the post title and post body using Visual Editor cells.
Viewing those on the front end should show just the post title and post content on an otherwise blank screen, but the widget areas from your theme were also being rendered in the footer area.
When a layout is assigned to a page it normally takes over the rendering of the whole page, but your theme is also inserting content on the page.
The integration with Genesis is difficult because it is not so much a theme as a framework, and child themes are at liberty to introduce all kinds of customisations which we cannot anticipate and which is why we can only maintain compatibility with the Genesis theme itself rather than its myriad child themes.
The child theme does appear to be the culprit in this case. I switched the theme to the parent Genesis theme and then my test pages rendered as expected, with just the post title and content on a blank screen.
I was unable to take a working copy of your site for local testing, but I did take a copy of your child theme and I have passed it to our compatibility team. If they are able to identify the problem I will let you know, as and when I hear back from them.
