I use the WordPress Screenshot Service with Link Essentials but it seems that no screenshots are prepared and saved?

The WordPress Screenshot service features a very restrictive policy for providing screenshots. If you use them to much, they refuse the service for some time. We recommend to use Page2Images which provides 100 free screenshots a day, if you register at their account. Also additional screenshots are very affordable.

If you have the need for an extremly high volume of screenshots, please consider installing wkhtmltoimage at your server to catch your screenshots yourself.