‘Easy WP SMTP’ plugin checks ‘manage_options’ capability in 2 places:
1) when adds own settings page under the “Settings” menu;
2) when add a couple of internal hooks for the wp_admin.
It does not send email messages itself. It just allows messages sent by WordPress itself to be sent via external mail server (SMTP).
You need to check why in some cases email is not built by WordPress itself or responsible plugin.