Are clothes stained by semen or urine unadulterated and pure – tahir – after being
washed in a fully automatic washing machine? In other words, do jurists consider that the various washing and rinsing cycles, in which water is repeatedly replaced, breaks the flow of
the water and so prevents it from being accepted as constantly flowing water – jari – which is necessary if things are to be rendered tahir?

There is no need for washing to be done in currently flowing water. The process described
in your question is valid for rendering soiled clothes tahir.
It is obvious that before you regard those clothes to be tahir, no pollutant – ayn al-najasah – remains on them.