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<?php
/**
 * I like better this filter, because allow access to the customizer,
 * but only to the desired capabilities given to the role.
 */

add_filter('user_has_cap', function($allcaps, $_, $args, $user) {
    if(in_array('customize', $args) && in_array('custom_role', $user->roles)) {
        return array_merge(['edit_theme_options' => true], $allcaps);
    }
    return $allcaps;
}, 13, 4);

<?php

/**
 * Old school, not really nice because first gives access to
 * a plethora of settings, then hides menus but they are
 * available through knowing the script: i.e.: widgets.php
 */

// add capability edit_theme_options to user

/**
 * Removes widget, nav_menus and AMP_customizer panels
 */
add_filter('customize_loaded_components', function($components) {
    $user = wp_get_current_user();

    if(in_array('custom_role', $user->roles)) {
        add_action('customize_controls_init', function() {
            global $wp_customize;
            $wp_customize->remove_panel('amp_panel');
        });
        return array_filter($components, function($component) {
            return !in_array($component, ['widgets', 'nav_menus']);
        });
    }
    return $components;
});

add_action('admin_head', function(){
    $user = wp_get_current_user();

    if(in_array('custom_role', $user->roles)) {
        remove_submenu_page('themes.php', 'themes.php');
        remove_submenu_page('themes.php', 'widgets.php');
        remove_submenu_page('themes.php', 'nav-menus.php');
        global $submenu;
        foreach($submenu['themes.php'] as $index => $slug) {
            if(strpos($slug[2], 'amp_panel')) {
                unset($submenu['themes.php'][$index]);
            }
        }
    }
});