I think this is right, but not especially important. You can have secular weddings and just go have the clerks sign the damn papers (I cannot claim complete knowledge of officiant laws in all 50 states). Again, I think it's correct to not privilege religion, and certainly specific religions, in this way, but it mostly falls in the symbolic category. I've officiated one wedding (had to write a letter to the office of Governor Patrick asking permission), and been a signed witness to a Pennsylvania "Quaker wedding" (no officiant, no need to actually be Quaker). So, yes, don't privilege "clergy" in this way but it's only a marginal issue.
