Has there ever been a published experiment to take a viral "census" of a human subject's blood?
You wouldn’t try to isolate viruses, this being the 21st century.  You’d perform metagenomics analysis, which is much more sensitive. 

That’s been done several times.  Two examples:


  Anellovirus and pegivirus C (GBV‐C) were detected among these donors. None of them were found solely in donors with abnormal liver enzyme. Anellovirus was highly prevalent (93.3%) and the co‐infection with multiple genera (alpha, beta, and gammatorquevirus) were more common in the donors with normal ALT values in comparison to those with elevated ALT ...


—-Full annotation of serum virome in Chinese blood donors with elevated alanine aminotransferase levels


  Anelloviruses were detected in each of the 10 plasma pools. Human pegivirus A (HPgV; GB virus type C) sequences were identified in eight of the 10 pools, more than 90% of which belong to Genotype 2. The recently described human HPgV2 in Flaviviridae was not detected. A small number of sequence reads of human papillomavirus were also detected in three pools. In one pool, two different gemycircularvirus genomes were identified and fully sequenced. 


—-Viral nucleic acids in human plasma pools