I would have thought this question is well beyond the scope of non-EU Europe, why not the actual Space Thread?
That aside, a 'Space Force' seems to require an opponent, whether that's a rogue (ostensibly?) Terran mission that needs to be faced off against or some actual incoming threat for which a pan-national response seems necessary. (So the US inaugerated one, in an invidualistic act driven more by personality than the (not entirely vapourware) threat from certain non-allied space programmes. It remains to be seen if that leads to Starship Troopers, de-orbitting response troops, Star Wars (in either Lucas or Reagan sense), Iron Skies or Iron Dome, above and beyond the mere consolidation of MilTech space assets out of the various original hands of the USAF, NASA and the like.)
If we're not going all "Space NATO" (and/or "Cosmic Warsaw Pact"), and perhaps even if we are, it should be possibly considered better as a UN(TLA) construct, held under the most global non-private umbrella there is, least it actually becomes a dish best served as Cold War, and it could be a very Cold War in space...
And although inevitably we'll one day have private/corporate force-projection (if we don't already, albeit not so overtly), I don't think we should attempt to keep it arms-length from individual or bloccy nations by this means either.
I have no real problem with the likes of ESA cooperating or partnering with non-ESA countries' existing or embryonic space agencies. There's ESA components going to Venus on NASA probes and NASA ones on the ESA mission, I think I saw recently. Though Roscomos might be looking to pull out of the ISS, whether or not they link up with the Chinese effort, they were a welcome enough part of that particular alliance of effort (including a significant time being the sole provider of man-rated rocketry).
For nominally civilian things, there are going to be various amounts of bed-sharing, and I'm only inclined to worry about it when (re)militarisation of space is a thing. Either that we're getting it, or that there's actually a frighteningly good reason for it.
There's a few points in your post I haven't really addressed (not sure if I understand the inspiration for them), and maybe I've drifted from the other points I thought I was addressing. But consider this my first thoughts to something that I'm only partly confident I understand enough to reply to.