What you are describing is one part of the immune response to most intact disease pathogens infecting cells.
If Covid-19 virus proteins are expressed by cells as a result of infection and ongoing disease, and those cells are targeted and destroyed by T-cells, why is this not also the beginning of autoimmune disease? And then why doesn't every pathogen triggering T-cell phagocytosis cause autoimmune disease?
The mRNA and DNA in vaccines are not stable, and are only transiently expressed for a matter of days, unlike replicating viruses, which can persist for weeks, multiplying thousands and millions of fold.
Why do you then think that this transient expression is more likely to produce autoimmune responses than infection by the actual pathogen itself?