Other way around, actually, in many cases. Witness the people saying myocarditis, blood clots, etc. weren’t happening prior to the vaccine, even though it of course one of the most widely reported concerns about COVID19 infections and the most deadly aspects of it. We had a full year that occurring prior to any vaccines being administered and people still made that claim. There have indeed been side effects some have experienced from taking vaccines - some very serious and even deadly - but those who then lied and claimed the serious and deadly symptoms of the COVID19 infections never existed prior to the vaccines in effect diminish this.
Yes it’s the spike protein itself found in both the virus and vaccine which is responsible for the vast majority of these issues, but the vaxx is worse because of the excipients it contains i.e. the lipid nanoparticle carrier allows the spike protein to more wildly spread to areas of the body beyond where it goes from a viral infection alone. And these excipients also have been shown to cause body wide inflammation along with other negative effects.