The main problem with Blue Origin is it operates a glacial speed. It was formed more than 18 months before SpaceX yet they have still not managed to fly anything into orbit. SpaceX achieved their first orbital flight more than 12 years ago. I'm sure Blue does have a lot of innovation, but they need to change their motto from Gradatim Ferociter to Gradatim VelociterBlue Origin is very innovative for sure. Right now they are focusing on making sub orbital spaceflights with "New Shepard" but when they get their "New Glenn " rocket up and running they should be right in there with Space X in commercial spaceflight.
I would advise caution concerning the difference between public and private companies*. Companies like Boeing and Lockheed Martin are large multinational corporations that are traded on the stock market. Their shares are owned, bought and sold by a vast range of private individuals, pension funds and many other organisations. SpaceX is a private company its shares are not traded on the stock market and it is owned owned by a select few people and organisations. I believe Elon Musk has controlling interest in the company.If you type in Space Company's, you will get an entire list of them. And an explanation of what it is they do. 95% of them are private company's? Only two of them are government owned. NASA and Roscosmos which is Russian.
They do work together. That is NASA's job, to make that happen. Apollo was done that way and now Artemis be being does as a joint effort using many contractors in the US, Europe, Canada and Japan.Could you imagine what we could accomplish if all these space companies worked together on a project? It may come down to that one day. Heck, if the entire would be involved. Like in The Martian.