I didn't see the video, but I'd say that society has decided marriage does affect society, hence laws regulating it (requiring registration of marriage with the state, for instance).
I agree whether or not my marriage is valid is of little direct personal concern to most people. But whether or not their marriage is valid is equally of little direct personal concern to me. In fact, that may be an argument for the state getting out of the marriage business. But the state routinely makes and enforces laws that affect individuals.
I'd also argue whether or not my marriage is valid affects nearly everyone I have a contract with, a loan from, am employed with, who may insure me, etc. So it turns out to be a lot of people. It's relevant to the Catholic hospital who must decide whether law requires them to grant me visitation, should my wife be sick. It's relevant to copyright law (copyrights typically transfer to spouses on death of a spouse). It's VERY wide-ranging. It affects whether I can buy $100,000 stuff on credit cards and then sell everything to my wife for $1 and immediately declare bankruptcy without paying any of the credit card bills. Etc.
For the US, the fire department example is a very bad one BTW. We have areas without publicly-funded fire departments in the US, where fire response is a private concern contracted between land owners and the fire company. As for government-run healthcare, I'd argue that communicable diseases are the reasons every state and most cities have a health department - clearly government involvement in health was considered important for other members of the community.