You are completely correct that we can formalize any Turing Machine as a FSM but! this _must_ include input. Again, this is just pure simulation as you state.
Again, we can write a _single_ Turing Machine program with a _single_ input baked in as an FSM and that input _must_ be finite. But that FSM is not capable of handling _any_ input like the the Turing Machine it is simulating. That’s kinda the big difference between Turing machines and machines of a lower class. Any FSM has limited input thus limited states, but a Turing Machine can run infinitely with finite input.
So, one can simulate the Turing Machine _with_ input using a FSM up to state N. But the actual Turing Machine can execute up to N+1 without having to hard-code any inputs and is capable of handling other inputs.
Again, your computer is a Turing Machine capable of accepting Turing Machines as input. We can hard-code any individual programs and make them into a physical computer chip — just as we did with our computers which can accept any input it can handle, limited only by its physical limitations. But these physical limitations are not an indicator of it being of a lower class because we can keep pushing those limitations back and still compute more of states of these programs.
Anyway, I’m glad you mentioned all of this because I went and pulled out my books.