Non-super-admin users cannot access CPT even though I have explicitly added the capabilities to the user role

Here’s how I’d handle it (mostly you’re right, and the tweak in your answer to use init gets you closer, but — as you’ve discovered — doing a bunch of switch_to_blog() / restore_current_blog() calls on every single page load is costly).

function add_opportunities_capability_to_admins() {

    // Set up the needed capabilities.
    $capabilities = array(
        'edit_opportunity',
        'read_opportunity',
        'delete_opportunity',
        'edit_opportunities',
        'read_opportunities',
        'delete_opportunities',
        'edit_others_opportunities',
        'publish_opportunities',
        'read_private_opportunities',
        'delete_private_opportunities',
        'delete_published_opportunities',
        'delete_others_opportunities',
        'edit_private_opportunities',
        'edit_published_opportunities',
    );
    $sites = get_sites();

    // Gets the current state of affairs.
    $current_permission_state = md5( json_encode( array( $capabilities, $sites ) ) );
    $last_saved_permission_state = get_site_option( 'my_permission_state', '' );

    // If things have changed, we re-run the permission updates.
    if ( $current_permission_state !== $last_saved_permission_state ) {

        foreach ($sites as $site) {
            switch_to_blog($site->blog_id);

            $admins = get_users(array('role' => 'administrator', 'exclude' => array(get_current_user_id())));

            foreach ($admins as $admin) {
                if (!is_super_admin($admin->ID)) {
                    foreach ( $capabilities as $capability ) {
                        $admin->add_cap( $capability );
                    }
                }
            }

            restore_current_blog();
        }

        // Save the current state so we don't have to run this again for a while.
        update_site_option( 'my_permission_state', $current_permission_state );
    }
}

add_action('init', 'add_opportunities_capability_to_admins');

Note: This code is untested. Try it on a test site before putting it into production.

References

• get_site_option()
• update_site_option()