This change is one that has been suggested by MANY people.
Although it would avoid the problem of the state forcing compliance on the Church, it would be more likely to end up with something like they have in Europe (which does things this way, for the most part) - most couples just use the state-sponsored method of getting official recognition of their marriage. Few bother with the Church Wedding (or, do it for the sake of the parents, who might be pressuring the young couple).
One of the side effects of Church weddings is that it's an opportunity for Church representatives to spend some time with the couple. It can be a teaching opportunity, and a point of catechesis.
I DON'T believe that this will lead to fewer problems - in fact, it is likely to lead to more. For those couples that were previously told that they couldn't be married in the Church, imagine the OUTRAGE of an already-married - or will-be-married couple who is informed that they are not eligible for a Church wedding. I can see the bonanza for lawyers who will sue the Church.
This will not just be gay couples, but also previously married and divorced people, cohabiters, and - if my expectations are realized - still-married potential polygamists.
The Progressive lobby will be encouraging the lawsuits, picketing, protests - not because they give a rat's @$$ about Church weddings, but because it will tie up Church funds, time, and energy, as well as provide a platform for their posturing.
Beware of "simple" solutions.