WordPress: Custom User Role cannot access Custom Post Type | “Sorry, you are not allowed to access this page”

Solution:1

You can use PublishPress Capabilities plugin to assign multiple capabilities to different Roles. You can download this plugin from here https://wordpress.org/plugins/capability-manager-enhanced/

Solution:2

With some playing around I realized I am definitely an idiot and WAY over-thought things. While I had previously read and tried some of the things in this similar post, I ended up substituting their code for mine and found it actually worked for my use case. In trying to understand why that was, I began trying to convert it to become mine and quickly found the root of my problem:

/* My Custom Post Type */
function custom_post_types(){
            register_post_type( 'staff', array(
                'labels' => array(
                    //labels redacted
                ),
                'has_archive'       => false,
                'hierarchical'      => true,
                'menu_icon'         => 'dashicons-groups',
                'capability_type'   => array('staff', 'staffs'),
                'map_meta_cap'      => true,
                'public'            => true,
/*---------> */ 'show_in_menu'      => false, /* <---------*/
                'rewrite'           => array( 'slug' => 'staff', 'with_front' => false ),
                'supports'          => array( 'title', 'thumbnail', 'custom-fields', 'revisions'),
                'show_in_rest'      => true,
                'taxonomies'        => array( 'member-type' ),
                'menu_position'     => 2,
            ));

In an effort to have a clean custom menu, I set show_in_menu to false which created the issues for me. When I changed it to 'show_in_menu' => true, my issue was resolved. In addressing this, I am tempted to just try remove_menu_page(); or perhaps consider something more elegant.

Anyways, the lesson for today is not to be hyper-focused on one aspect. Hopefully this helps someone else and happy coding!