As a technical point, viruses are only alive inside cells, as they have no machinery to replicate, repair, metabolize etc, they use cellular mechanisms. The thing is that noncytopathic viruses do not usually lyse the cell and release millions of copies of themselves, so blood levels of virus are very very low. Regular lytic lifecycle viruses make millions of copies, explode the cell, then get released into the blood. This is what you see on a blood viral titre test. I suspect though that if we developed tests for specific viral proteins they might be detectable even in noncytopathic infections, and also that we might have antibodies to those proteins.
If the immune system is responding properly you should also see lots of antibodies to the virus from about week two of the acute infection, or week one if you have had that virus before. Those antibodies are what is measured on an antibody titre test. The antibody tests are indirect, there is no way to be sure they are to exactly the virus you are looking for (they are only semi-specific) and if anything is wrong with antibody production, which appears to be the case at least some of the time in ME, then antibody numbers might be lower than they should be, or even nonexistent. Many of us lose the ability to produce antibodies to prior infections - something is wrong with the B cells. We are still trying to figure this out.
Also it is my understanding that for enteroviruses there are two different noncytopathic forms currently known ... so you could wind up with three infections from the initial enterviral trigger, one acute infection that should resolve fast, and two smoldering infections that appear to be for life unless they can be treated.
The only way to be sure if noncytopathic viruses are cleared is not by any sort of blood test, but by biopsy.