> The Russians **have had** their favorite, most-hated pindoses.

The author uses the Past Perfect to reflect the fact that at different periods in time, different "pindoses" featured prominently in the Russian public opinion. 

There exists a long series of US persons who Russians loved to hate. This tendency to hate some US persons continues up to the present moment, and could continue further. At each subsequent period in time, some of these US persons were "most hated". As time progressed, some of these "favorites" drifted out of public view, and others assumed their place. 

If we change the tenset to Present Simple, getting

> The Russians **have** their favorite, most-hated pindoses.

The sentence would mostly refer to the set of most hated US persons that exists in the public mind at this moment. Some of these most hated persons might be historical figures from decades or centures ago, some might be figures still alive, but this set won't include figures that have drifted out of the public view by this moment.