Beyond that, the virus causes orders of magnitude more spike protein response, so even if that was true, which it probably isn't, then it is still very unlikely that the vaccine could cause long-covid.
Also, all vaccines introduce spike proteins in the body, not just mRNA vaccines.
https://www.sciencedaily.com/releases/2020/12/201217154046.h...
Relevant part "The spike proteins alone can cause brain fog". And vaccines do produce viral proteins as intended.
It appears to come from a researcher contacted by the site for the article. And it's probably true.
However, there's a few reasons to believe that "long" covid like symptoms wouldn't come from the spike protein alone:
1. As a parent mentioned, you generate a lot fewer proteins w/ the vaccine than the virus
2. IIUC, those proteins are more localized. The soreness with an initial injection is inflammation due to spike protein creation, and that's usually localized to the injection site. The proteins themselves are less likely to travel as widely as the virus.
3. The proteins don't last very long. They're gone after a few days. So long term symptoms wouldn't be due to a continued immune response. They might be due to inflammation that hasn't gone down, I guess, but that seems unlikely to last months.
And again, all of these will be worse with the virus than the vaccine. You'll have more spike proteins, for longer, over a larger part of your body, and also have a virus attacking you in addition to the immune response.
There aren't actually any virus particles in the mRNA vaccines, so there's no biological mechanism for that to happen.