I think your bird analogy is a bit much, as I've never seen a bird trying to access their 'loved one' in the hospital.
Marriage in its applications is a contract of varying components, including economic, social, and others as desired. Unfortunately, it is up to the government to issue this contract, instead of just to enforce them as you would want from a standard minarchist contract law arrangement. This means the government controls who is allowed to seek what social or economic commitments from whom, and so forth.
In short, government involvement is generally bad, but the degree of government involvement is a matter of discussion. Moving the matter to the states is one aspect, but removing it from government entirely except for contract enforcement is probably the best solution.