I mean - in a shooting war on earth, one thing you could do that would have an immense impact on everything else would be taking out the oppositions satellites.
It would effect the oppositions communications, their reconnaissance, their targeting via GPS, their ability to navigate. You'd have a HUGE advantage - and would also be immensely disadvantaged if the opposition took out your satellites.
And that's assuming that there are no satellite based missiles or other weapons.
I think I'd just call it part of the air force, but I'm OK with it being the space force instead. I know in some ways it has become a little bit of a joke - but there is some reality behind needing to defend US interests in space. Or at least on the edge of space.