Without profit as a motive, innovation would be decades behind (if not longer than this). Governments can barely afford crumbling infrastructure maintenance as it is. I seriously doubt they are going to invest in projects for the 'greater good'.
"We've ran high-risk R&D projects successfully before as public projects - Manhattan project, the space race & moon landing, ARPANET, etc."
Yes, for defensive or offensive military purposes. Not much beyond this.
Even big pharma supplies the world. The rest of the world with socialized medicine create knock-offs at a fraction of the cost, because they didn't have to spend decades going through testing and billions of dollars developing it.