A minister is not an agent of the state. He is not a civil servant. A minister performs a wedding ceremony only because he represents the church, as one called out to the teaching office in the church, with the church gathered as witnesses to the vows being made.
In the case of unbelievers, a minister of the state (such as a Justice of the Peace) is perfectly appropriate to officiate because it is the state, not the church, that will hold the couple accountable for any breaking of the vows made.
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