You still are looking at this backward.
The church cares about civil permissions and authority. Even in states where marriage is purely a spiritual thing they first require a civil marriage.
The priest does this because they agree to terms and conditions. The terms and conditions of having the state do the legwork to find the documents regarding identity is that a priest must adhere to this under all circumstances.
You seem to have this idea that the church is held captive. It is not. It is using the simpler, less onerous way of doing things and thus AGREES to do this in all circumstances. It’s a basic case of “you abide by my rules, I’ll abide by yours”. Both parties agree to accept the consequences if they do not behave accordingly. In America, this means that ministers cannot be forced to marry two persons for any reason they choose—immaturity, age, gender…etc. The state cannot force a minister to perform. The minister accepts that he must use the state to gather documents. It’s a mutual agreement for the benefit of both.
Should this change, I’m sure the Church will, too. Even if the church moved to sacramental marriage in the countries where this is the case, they STILL use the state as a means-test for identity and validity. Should the state ever become unreliable in that the Church would still require civil documentation.
Which, I suppose in your view, would still be them “giving” authority. Which is backward because the church has the duty to protect the sacraments and establishing basic identity is key in that.