You miss the point of space based weapons. Space is big. Really big. You take advantage of that space. You wouldn't put them in LEO. That would be stupid. You would put them way-way out, so that there is little chance you can preemptively eliminate them. This is why we wanted a nuke base on the moon in the 60s. Strategically its the most survivable deterrent.
You are correct, space is big. But LEO or as close to that as you can be is critical. The reason ICBMs were so popular, is the speed that they have, and the lack of reaction time that gives your opponent. If bombers take six hours to get their payload to Moscow, then an ICBM that can get there in twenty minutes is a vast improvement. LEO basically cuts that time in half or more, since there is only the coming down part.
Yeah, the Moon is far out of reach of aircraft launched ASAT weapons, but it is also three days away. Old propeller driven aircraft is faster than that, by twelve times. Days gives your enemy plenty of time to get their anti-missile weapons platform carefully lined up, and perhaps time to even have secondary measures prepared. Entire cities can be evacuated in three days. Sub pens and silos can be emptied, and divisions can be dispersed.
This of course ignores how incredibly fragile nukes and rockets are, and how space is incredibly expensive and dangerous to be in. One solar flare or fleck of paint can reduce your billion dollar weapon platform to junk. Space is a stupid place to put nukes, and LEO is the only place they would be 'fast' enough to be scary.
Want another reason not do space weapons at all? There is zero stealth in space. You cannot hide shit up there. We can see things in thermal really easily against the cold background. Anything man-made will stick out like a sore thumb because of its heat signature. Everyone with a cheap telescope will be able to watch everything you do on your moon base, unless you put it on the far side of the Moon, and that creates a whole new set of problems.
This space force is yet another idea from an inept imbecile who isn't listening to his advisors. We could have put nukes in orbit anytime in the last sixty years, and yet, somehow we chose not to. We had plenty of hawks in the oval office during that time, and yet not one of them even suggested such an idea. Why is that? Old Ronnie was a commie hating hawk, and even he only suggested the Star Wars Defense Initiative as an anti-missile deterrent. Far smarter presidents (from both sides of the aisle) have passed on the idea, for very good reasons.
The Air Force traditionally has been the branch that has had responsibility for space related weapon platforms (ASAT, spy sats, high altitude recon, etc), so why would you create a whole new branch for this? I would wager if I had high enough security clearance, I could find an existing AF officer in charge of super secret space based weapon research projects that are already underway. This is just pointless redundancy in creating a new branch. "Drain the Swamp", indeed!