Yes they do care, but you know what? The ones who really care, really really care do not marry in the first place because like it or not, sooner or later somebody will call them Mrs.
You are part of society, by marrying you are declaring to society that you wish to be known as a couple, you are getting certain privileges for this status.
I believe married couples don't pay inheritance tax? Something like that.
Then you complain that society changes the title it's given to married women for ages to accommodate your preferential title.
Somebody making the terrible mistake of calling you 'Mrs' is some kind of terrible affront. Pass the smelling salts.
Your relationship is no longer a private matter. That ship sailed when you married.
And all this snooty stuff from some posters about it being just a legal contract, if it's just that, why have anybody else there apart from the two witnesses?
Ultimately you want the best of both both worlds: be recognised by society as a couple AND that society to bow down to your (niche) expectation of not being called Mrs when it's the title most women who are married are called!
And if you really wanted to fight the patriarchy, you wouldn't get married in the first place.