But not once have they actually bothered to examine or even question the legitimacy of the PCR testing methodology. It got touted as the 'gold standard', case closed.
PCR testing is the gold standard because of how specific and powerful it is at amplifying DNA. A set of primers to a gene will allow you to accurately copy it from truly tiny samples. They need picograms or less of viable DNA to amplify successfully.
Specificity of pcrs is also extreme, producing separatable bands on an agarose gel even when amplifying off of a more complex organism's whole genome.
The company idt sells primers that recognize the nucleocapsid coding region of the genome as well as plasmids (circular DNA) containng that gene. The genome of covid is openly available, and these primers clearly bind to the region they are supposed to (you can match their sequences up on a computer).
PCR primers also amplify extremely specific regions of the genome. Each base in the primer gives a 4 to the n increase in specificity of the sequence, so a short 20 base primer will only to 1 in a trillion sequences. This is why pcrs can be used to get dna from sewege plants and archaeological findings; contamination of samples with other DNA is not an issue.
I am not sure how you think pcr amplification is not legitimate. It is probably the most accurate and sensitive way to identify the presence of an organism. Furthermore, the methodology is transparent (pretty much any lab with a pcr can run these tests. I know a lab that bought primer sets for in house testing when it was more backlogged - if there was some conspiracy with it giving false positives, people would know.
Also Chinup is accurate about sequence alignments. The human genome was sequenced through shotgun sequencing. This is how genomes are sequenced typically because you get a much higher accuracy (alligning fragments gives you multiple references for each spot in the genome making it a very robust technique).
Unfortunately simplicity is not a requirement of truth.