That may be true overall (and it may not btw, only time will tell), but it's certainly not true on a by-indvidual basis. Someone who's, say, 18 with no major health disorders is going to experience a lot more risk of adverse outcome from any of the vaccines than from SARS-CoV-2. That's partially a function of the incredible SARS-CoV-2 outcomes in the "not old" age range but also because vaccination is also a fundamentally different intervention than exposure to the entire SARS-CoV-2 virus. If we're talking about the mRNA vaccines (Pfizer/moderna/etc), those are basically spike protein mRNA suspended in lipid nanoparticles. It makes your cells express the spike protein, not the whole SARS-CoV-2 virus. So metaphorically speaking you're getting pure spike protein antigen, and different peoples' genetic predispositions/etc are going to play out differently in that scenario as opposed to contact with the normal virus.
...Like those of the people in the case report we're looking at.