In theory, yes. In the real world that actually exists however, it would be very hard for a film to get released in many theaters without a MPAA rating. I don't see how the studios would be "forced" to use the government's rating if it did exist anyway. Like I said earlier, I don't think the ratings should lead to enforcements on how movies are show. The ratings would just exist on a government web site somewhere. If someone wanted an alternative system they would be free to make one themselves.
Theaters can refuse to use the MPAA if they want. Theaters will want to turn down the MPAA if their customers want it. The reason they don't usually is because customers are happy with the MPAA.
If the government's rating is non-binding, I guess that amounts to almost the same thing. I'd still trust them less, however, because they are not hurt if customers don't like their ratings. If no one uses the government rating, that agency will still get funded by the government, and thus by the taxpayers. They are still less accountable.
And in practice I don't know of any government agencies that operate on a voluntary basis like that. Even the ones that don't force you to listen to them (like the Post Office) have mandated monopolies.
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your 7 time champion, Link.