No, we're only paying for part of 'it'. The cure isn't just research, it's also logistics and there does not exist a capable non-profit to do the logistics part of it in such a short time. Also you underestimate how much it costs to build such a supply chain capable of producing and shipping that many vaccines around the world. Hint: the research costs are just a fraction of the logistics cost, you just don't see that up front because it's also used to ship other stuff besides vaccines.

These things cost money, and you assume your taxes are covering them in full. They don't.

I think the key desire is that the intellectual property, built from publicly funded research, should not be owned by a single corporation. Production and distribution are of tremendous value and these companies should be rewarded for providing that value. But the vaccine itself? Why not let others invest capital and try to produce & distribute the vaccine?