Category Search / Custom Post Type search on my website. Custom Post Types that are “page-like”?

You can limit a search to a particular post-type quite easily: https://www.wpbeginner.com/wp-tutorials/how-to-limit-search-results-for-specific-post-types-in-wordpress/

Or a nice simple way to limit a search to a particular category by adding a hidden field into the search form: https://www.wpexplorer.com/limit-wordpress-search/

In terms of post vs page for custom post types, you can find the differences on the documentation page for setting up custom post types. See “hierarchical” option. You’d probably want the page type (hierarchical) https://developer.wordpress.org/reference/functions/register_post_type/

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