You definitely need to set map_meta_cap to true instead of false to create custom capabilities.
I haven’t seen the 'capability_type' => ['note','notes'] way of creating capabilities before – maybe it’s shorthand, but if just changing map_meta_cap doesn’t work, you might want to spell everything out the long way:
<?php
$args = array(
'labels' => $labels,
'public' => false,
'show_ui' => true,
'show_in_menu' => true,
'query_var' => true,
'has_archive' => false,
'menu_position' => null,
// The most crucial change: true
'map_meta_cap' => true,
// Possibly required: spelling out every capability individually
'capabilities' => array(
'edit_post' => 'edit_note',
'read_post' => 'read_note',
'delete_post' => 'delete_note',
'create_posts' => 'create_notes',
'delete_posts' => 'delete_notes',
'delete_others_posts' => 'delete_others_notes',
'delete_private_posts' => 'delete_private_notes',
'delete_published_posts' => 'delete_published_notes',
'edit_posts' => 'edit_notes',
'edit_others_posts' => 'edit_others_notes',
'edit_private_posts' => 'edit_private_notes',
'edit_published_posts' => 'edit_published_notes',
'publish_posts' => 'publish_notes',
'read_private_posts' => 'read_private_notes'
),
'rewrite' => [ 'slug' => 'note', 'with_front' => false ],
'supports' => [ 'editor' ],
'menu_icon' => 'dashicons-format-aside',
);
register_post_type( 'note', $args );
?>
You may also want to grant your custom role a few other capabilities:
<?php
$role = get_role( 'my_custom_role' );
// Read (front end) all post types
$role->add_cap('read');
// Adjust their dashboard
$role->add_cap('edit_dashboard');
// Upload files
$role->add_cap('upload_files');
// See the list of users (but not manage them)
$role->add_cap('list_users');
// Allow taxonomy management - Categories and custom taxonomies
$role->add_cap('manage_categories');
// Use the Customizer
$role->add_cap('edit_theme_options');
// Only if you really need to, allow them to paste in HTML/JS
$role->add_cap('unfiltered_html');
?>
At a minimum I usually grant “read” to custom roles so they can see the front end of the site.