Thanks for pointing that out. Hopefully, I can help to clarify. This is quite a common practice when phasing out an assay from any lab supplier. The link that you have is to a lab notice of the discontinuation of the CDC 2019-Novel Coronavirus (2019-nCoV) Real-Time RT-PCR assay. Not because it has issues with the assay as it actually only looks for SARS-CoV-2 (2019-nCoV_N1/N2: targets virus nucleocapsid (N) gene for specific detection of SARS-CoV-2), and not influenza. The CDC is doing so because there are many commercially available assays on the market. Opting for a commercial-grade assay especially with higher throughputs than the original CDC assay will help labs if they haven't already switched. There are over 380 assays approved for COVID detection.
There may be confusion because the CDC 2019-Novel Coronavirus (2019-nCoV) Real-Time RT-PCR assay was designed to be used on existing RT-PCR testing equipment that is usually used for testing for the seasonal flu. Like the vaccines, this was done under the EUA to get testing ramped up and running early on in the pandemic.
Discontinuation of the assay has nothing to do with it detecting influenza as it is not designed to do so. This is merely market factors, newer and faster assays and it no longer needs to be produced. So they are discontinuing it.