I think you misunderstood the (G)GP comment which mentioned cross-reactivity, i.e. the immune system's antibodies cross-reacted with both the spike protein and an endogenous human protein, like what can happen with Epstein–Barr virus Nuclear Antigen-1.
The immune system doesn't generate antibodies to an entire virus. It generates antibodies to specific immunogenic proteins or parts of proteins.
If the spike protein resulting from a vaccine causes an immune response that generates antibodies to the spike protein and oops, also some endogenous human protein, that's autoimmune disease.
If you get infected by a virus with the same spike protein, your body will generate antibodies to the same spike protein and also cause autoimmune disease if the antibodies are cross-reactive.
If cells expressing exogenous immunogenic proteins triggered autoimmune disease[1], even the existing wild-type (alpha variant) vaccines would cause autoimmune disease. Presumably that's not happening. The immune system may attack those cells expressing the exogenous vaccine-coded spike proteins, but normal immune systems down-regulate it well enough that it doesn't cause major problems. It would be a sign of a broken immune system if it decided to initiate a long-term attack of human cells just because they once expressed an immunogenic protein. If it did that, any virus would cause autoimmune disease.
Please tell me you work in a field related to immunology and you have a subtler point that I'm missing, and that you didn't just create an account to criticize a point you misunderstood yourself.
[1] to an unacceptable degree. Obviously, things can go wrong with the immune system and almost anything could, if you're unlucky enough, trigger an immune reaction leading to autoimmune disease eventually.