Marriage is more analogous to a birthday. (A personal change in status)
Wedding is more analogous to a birthday party (i.e. the event celebrating the change in status).
As you pointed out in your logic, the birthday gift isn’t really about the birthday party, just like the ring doesn’t commemorate the wedding celebration, it commemorates your new marital status.
Unless of course you are the kind of person that is so focused on the wedding celebration that you forget the reason why you are celebrating to begin with (spoiler: you are making a commitment and entering a new life stage).