> Why is it always these efforts are little private things or not for profit initiatives?

Because (relatively) small teams can act faster. It is also a lot easier to have a small, motivated core group with shared objectives and aligned vision of a project. A non-profit/public benefit entity is also a reasonable host for such an effort. Such projects can also tap into public money (under particular circumstances) via the various EU and national grant schemes.

> Just have it be an official EU government backed identity.

The amount of baggage this would entail is quite large. Besides just bureaucracy you almost certainly would then also invite politics even more, because, even for small projects, the next question will be: In what countries is the service hosted and by which agency/developers?

That is besides the almost inevitable inefficiencies for resource usage and other obligations from being associated with public funds (which can require a LOT of internal politics work to acquire and especially retain).