That's okay, I'm the only person I know who's talked in support of the idea too. To me a guaranteed minimum income just... Makes sense to me. That's the closest we'll get to each person getting the opportunity to pursue happiness.
I'm way crazy liberal, but if everyone had the ability to take care of themselves, and live as they want, well, we wouldn't need a minimum wage. You're right. (But then, I'd want health treatment to be like police/fire. Something everyone needs, and we're better all paying for.)
So, yeah. The military, police, fire, health, roads, public school to about 10th grade, I guess the post office as we need a way to ensure public notices, and provide a baseline life through these means? I'd good with that. That's the only government I need.
Just about any other social institutions can be done by people living on their GMI and aided by donations.
I'm the only libertarian I know who is provisionally in favor of a universal income. But once you have that, you don't need a minimum wage and in fact you don't want one, because you want to maximize the number of people who can find work. The minimum wage exists to prevent exploitation, but if everybody automatically receives enough to live on, that's no longer a factor.