I agree with most of what you say. I heard the Vaccines President of GSK on Radio 5 earlier in the week and the main lesson from the pandemic for them is that they will run production facilities at around 40% in future (outside pandemic times) so that they can quickly scale up production to respond to pandemics. He said it's too difficult and time consuming to scale up from a standing start.
Our Government says it's about private companies but nations are important to vaccine development and roll out as discussed on here. For example, Oxford were steered to partner with AZ. Yes, the nature of contracts are very important and many of the companies are multi-nationals but politics are a factor and they often reflect national interests.