Very good post from someone (I presume) not in the medical field. You are making effort to read up and educate yourself, something I wish more people do. I just want to add couple of things.
First, scaling is a matter of resources and funding. We are the richest and most resourceful country in the world. If we have the will we should easily accomplish effective testing policy.
Second, the available tests you mentioned PCR and antibodies test, each has a completely different role. One is a "viral" test to detect if parts of the virus RNA is present denoting ACTIVE infection while the antibodies test is and indication of PREVIOUS infection. It has no role of determining whether someone is capable of infecting others. It could take 2-3 weeks after the the infection for the antibodies to be detected in the blood. Most patients are "viral negative" by then.
One more thing. In addition to the standard PCR test that was developed by CDC in early 2020, Quidel corporation (couple of miles from my home in San Diego) developed a very fast (15 minutes) viral test based on its Antigen Flourescent Immunassy technology that been used to test for original SARS Covid virus (pandemic novel SARS actual name is SARS Cov-2). This test has been validated and approved by the FDA and you get the result before you get back home from the test site. It is done on site at parking lots in Southern California and the specimen is NOT sent to a lab. I had this done twice already since I work with Covid-19 patients. The main issue with this test it is not cheap. If we have government subsidy and scaling this test is superior to PCR because of ease and speed. Also it can be done at doctors offices and urgent cares, etc.
https://www.medtechdive.com/news/qu...-test-is-now-on-par-with-pcr-accuracy/581902/