A government funded, unionwide research agency does sound like a good idea to me. But DARPA is a military r&d agency and I’d rather see an agency focused on benefitting civilian life, preserving nature, sustainability and such.
EU would clearly benefit from concentrating each nations military industries into unionwide companies and projects, too, though, but idk, I feel like we as a society could benefit from a strong, well funded civilian research effort.
I don’t know if there is such already out there, all I ever hear about is either underfunded or financed by private profit-oriented companies.

Sounds like a good idea, but it would probably be better to have a military and multiple civilian agencies specialized in different fields. Like medical, physics, chemistry and so forth research agencies. Also in edge cases both agencies should be allowed to do research or cooperate on it.

Having a government backed research agency might be good seeing as that has declined in recent years.