Yes a government license on marriage is needed. This is in order to legally keep track of records. Getting married is entering into a legal pact in this country and as such the government has a right to be involved in the process. They need to file the license for protection.
Without government licensing of marriage, common law marriages will be more and more undefined. Marriage licenses allow government entities to collect fees for revenue. Recognizing marriage also gives tax benefits, so there should be some legal document saying two people are married. A legal license for marriage also means there is no disputing what the legal relationship between two adults is.
The 10th Amendment states that non-enumerated rights fall to the states and the individual. Since states cant marry, this right falls onto the individual. So the individual has the right to marry and claim any government benefits that might yield by registering as married to obtain these, yes. The government however does NOT have the right to require this information from them though. I agree on your points that registering a marriage with the government has benefits, I disagree with the comment that the government has the right to require it.