Does that not suggest to you that maybe you were wrong? It's not just the decrease in testing, since test positivity has decreased from the peaks even as total test count plummeted.
There is no evidence that PCR tests greatly overcounted SARS-CoV-2 infections:
1/2. The specificity of commonly-used PCR tests for SARS-CoV-2 is clearly very high, >99.9%, since there have been regions (e.g. Australia) where total positivity was <0.1% for months at a time. It's true that other PCR tests (like your whooping cough tests) may have much worse specificity, especially in the early days before primer design was as well-understood; but we know empirically that these ones doesn't.
Sloppy lab work could result in false positives by cross-contamination, but contamination has to come from a true positive sample; so this is a problem only when the true positivity is high. Sloppy clerical work could result in any possible incorrect result, but you've presented no evidence that this is a significant problem in aggregate.
Lots of papers studied the relationship between PCR positivity and culturable virus, for example
https://academic.oup.com/cid/article/72/11/e921/5912603?logi...
It's true that at high Ct, the cultures are often negative; but (a) the relationship between Ct and culturable virus may vary dramatically with small changes in PCR protocol, (b) a patient early in their infection may have high Ct now but lower Ct later, and (c) it's not clear whether all naturally infectious patients would be positive by artificial culture anyways. Based on these factors, most public health authorities chose not to use an artificially lower Ct. So a positive PCR test doesn't necessarily mean that a patient is infectious now, but it does mean that they are or were recently (weeks to months) infected.
For emphasis, it is absolutely expected that some samples will be positive by PCR but negative by cell culture--the PCR is testing for any RNA matching the primers, but the culture is testing for replication-competent whole virus. Presence of the latter implies the former, but the converse isn't true. The public messaging on that wasn't very good, and perhaps they initially misinformed you; but no one with any background in molecular biology was confused here.
3. Your idea that the PCR test is simply assumed to be accurate is completely false. The point of papers like the one I linked above is to compare two test methods, find the places where they disagree, and try to judge--from the experimental results and from our understanding of the biochemistry of the test methods--the reasons for those disagreements. They're not assuming either method to be perfect. Based on many such comparisons--against cell culture, against electron microscopy, against antibody tests, against excess mortality once that got big enough to distinguish statistically (which it mostly wasn't by Jul 2020), etc.--public health authorities judged PCR to be the best available gold standard, and good enough for many statistical purposes.
The public messaging was again pretty bad there, and if you looked only to that then I understand why you'd feel misled. The actual scientific literature is there for anyone who wishes to read it, though.