Implicit in the popular, but not necessarily legal, meaning of the word “marriage” is intimacy or a very special personal relationship.
A very special personal relationship may or may not include intimacy, so I think that is not germane to the point.
You and I apparently agree that there is nothing about "intimacy" implied by the legal institution of marriage, so I am still left wondering what led you to observe that religious conservatives and the gay community want government involvement in their intimate personal lives.
It appears to me more a case that the religious right wants to create a standard for marriage relationships between same-gendered couples that does not exist for different-gendered couples; namely, that same-gendered couples are
assumed to have intimate sexual relations that the religious right believes to be "an abomination" and thus should be prohibited by law from the legal institution of marriage, whereas different-gendered couples are
not assumed to have any sexual relations at all.
Does that seem a fair assessment to you?