The Search module from CMS core allows to limit the Content type and is quite suitable for use, the settings are available for understanding. The form is added to the pages like other blocks /...
I played around with metatags and when i revisited those pages they now have description metatags. Not sure what the initial problems were though. Sorry for the time waste.
Layout is not Content, it is not used to create pages, so I think you have chosen the wrong way. To avoid problems, just need to use every tool for its own purpose. This is not a direct answer,...
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This issue may be related (and it looks like it has a PR that can be tested):
https://github.com/backdrop/backdrop-issues/issues/2134
That PR wont be in production any time soon. But you could make a quick access condition module, its quite simple. Use as a guide: https://github.com/backdrop/backdrop/blob/1.x/core/modules/layout/plugins/access/path_layout_access.inc or https://github.com/backdrop-contrib/block_scheduler
Your form function would probably be something like:
And your access function:
Some adjustment may be necessary.
As an easy workaround without any coding Views could be your friend.
We'd need some more info about your use-case though. Could you describe a bit more, please?