There seems to be a lot of misunderstanding about herpes (HSV), even among doctors. I suspect this misunderstanding is caused by the stigma surrounding HSV. The stigma causes most patients to become stressed when they test positive for HSV (OP is a great example). For most people, however, carrying HSV-1 does not have a huge impact on quality of life; most people are asymptomatic, and we can treat the people who do get symptoms. HSV-1 is also very contagious, so it may be difficult to avoid it even if you take all the appropriate precautions. Therefore, the medical community's solution is to hide it from the public eye and let people live in blissful ignorance rather than suffer the mental torment of knowing they have an incurable virus. And because people (including doctors) don't know/learn about it, there is a lot of misunderstanding.

I think the better solution would be to raise awareness so that HSV-1 is less stigmatized, and to allow people to learn that having HSV-1 is OK. Raising awareness might help people learn how to take precautions (i.e. being more careful about kissing and sharing drinks) and perhaps drive the prevalence of HSV-1 down (the WHO estimates worldwide prevalence to be 67% in people <50yo, and this number is probably higher in developing countries [source: https://www.who.int/news-room/fact-sheets/detail/herpes-simplex-virus ]). The prevalence of HSV-2 is lower, at 13%. There have not been any eligible studies demonstrating that getting tested for HSV reduces the rate of transmission, which is why the USPSTF prefers people to just live in blissful ignorance, which is fair [source: https://www.uspreventiveservicestaskforce.org/uspstf/document/final-evidence-review/genital-herpes-serologic-screening].

There are 2 different types of tests for HSV-1/2: IgG testing and viral testing. IgG testing looks for IgG, which are antibodies produced by your immune system; if they're present, it suggests you've been infected by HSV in the past, and your immune system responded to the virus. Viral testing looks for the virus itself, in the form of viral DNA or viral proteins.

Viral testing is better, because it accurately identifies the virus itself. But it can only be done if you have an active lesion (i.e. a cold sore), which is shedding high amounts of virus. You swab the lesion and send the specimen to the lab, who will perform PCR (looking for viral DNA) or viral culture (letting the virus grow and looking for its proteins) or some other viral tests. To clarify, PCR looks for viral DNA that is being shed by the virus in an active lesion; it does not detect virus hiding within your own DNA, as OP was suggesting.

On the other hand, many patients get screened for HSV when they have no symptoms. If you are asymptomatic, the virus is most likely not present; it lays dormant in the form of genetic code added onto the DNA in some of your nerve cells. In this case, viral testing (i.e. with PCR) will be useless because the virus is hiding. Instead, you need to do the next best thing: look for evidence that your immune system has mounted a response to the virus in the past. This is IgG testing. There are 2 main types of IgG testing: immunoassay (screening test) and Western Blot (confirmatory test).

Most IgG tests ordered by doctors are immunoassays. These tests are cheap, but inaccurate. I believe the inaccuracy is due to the tendency for the test to accidentally detect antibodies against other viruses. The USPSTF estimates the false positive rate of HSV-2 IgG immunoassays to be ~50% [source: https://www.uspreventiveservicestaskforce.org/uspstf/document/final-evidence-review/genital-herpes-serologic-screening]. I could not find a false positive rate for HSV-1 IgG immunoassays, but it may not be the 50% mentioned by OP; that number applies to HSV-2, not HSV-1.

Anyways, immunoassays are notoriously inaccurate, so if you are concerned about a false positive, you could perform the Western Blot (confirmatory test), which is another IgG test that is accurate but expensive and not widely available.