On the larger issue, surely we should leave it to the state to register marriages. There is no demand for organisations to become registrars of deaths or births (unless I'm greatly mistaken).
We already have that as far as I understand it. For civil ceremonies we have HSE registrars (who don't work on a Saturday) and religious bodies. The religious body (i.e. the priest) is serving two functions. He's marrying people in the church in a religious ceremony and he is separately registering the marriage on behalf of the state as a registrar.
In any religious weddings I have been at there have been two separate things happening. The exchange of vows is the religious element and separately the register is signed by the couple and their witnesses in front of the priest. I could be wrong, but if the second part is not done, the couple may be married in the eyes of the religious organisation but not in the eyes of the state. For example, in the case of a Catholic marriage where one of the couple has had their first amrriage annulled, the state doesn't recognise the marriage and so no registration would take place.
I wouldn't begrudge anyone wanting to have a religious ceremony and be able to do the legal/ state bit at the same time. Nor do I have a problem with priests being allowed to carry out this function on behalf of the state. It has no effect on my life, nor does it on anyone who doesn't want a religious ceremony.
This Bill allows people to have a non-religious/ humanist ceremony and not do the legal bit sepearately to have it recognised by the state. It also now allows people to get married on a Saturday and do it all together as religious ceremonies can.