If they did that, wouldn't they need to change the process? Didn't the COVID vaccine originate from a fairly new method of understanding the virus, meaning it actually can somewhat quickly react to change vs. the prediction for the flu vaccine?
Pulling in a few threads - and please keep in mind that I'm not a scientist nor a clinical person, this is all from what I've read, what's been explained to me by clinical folks that I work with and/or trust, and that I've used to help design workflows for the healthcare system I work for. Do not substitute actual clinical information and advice for the stuff I say. See your doctor.
The Moderna and Pfizer covid vaccines are mRNA-based, which is different from previous vaccine technologies.
https://www.wedovaccines.com/about-vaccines#how - some helpful info by Novavax. But there are multiple types of vaccines in use today, there aren't "covid is different and all the rest are the same" that you hear from the derps. There have been a
lot of advancements in vaccine research and technology over the years. There are multiple "types". Multiple ways to administer.
There are also combination vaccines like the MMR, which blends vaccines for measles, mumps and rubella (hence the name). I hope that we can, eventually, wrap seasonal vaccines into one mRNA "shot", where Covid, influenza and possibly others can be administered all at once. I do think we'll get there, but not soon.
And yes, to whomever mentioned that the CDC is the main developer of the flu shot, and then other companies manufacture it. Absolutely correct. But no one knows or cares about that like they do with Covid.
edit to add: all of this is even more to say that the right wing anti-vax movement is 100% counter-factual and is aggressively ignorant and/or purposefully misleading.