The government already does not interfere with the private, religious aspects of marriage. The Catholic Church can define marriage however it wants, and the state has no say in it, which is why you can't get a same-sex marriage in a Catholic church.
What's being debated is who should be recognized as married for official purposes (5th amendment rights, tax filing, survivor benefits, inheritance in the absence of a will, etc.). On that question, Amash supports the Defense Of Marriage Act, which says that the government should recognize only opposite-sex marriages, and not recognize same-sex marriages. If he opposed DOMA, and instead introduced legislation to abolish all government recognition of marriage, I'd give him kudos for that!
It shouldn't make a difference whether or not someone is married... you know, equal protection under the law. The solution to things like survivor benefits and the others you mentioned (and you're right, Amash doesn't mention this solution) is to simply have private contracts that people decide upon themselves.
"If he opposed DOMA, and instead introduced legislation to abolish all government recognition of marriage, I'd give him kudos for that!"