I would recommend that folks check out the example of the Purple Line. That is a light rail project in suburban Maryland outside of Washington, D.C., where a public-private partnership was expected to provide construction and 30 years of operation. That partnership collapsed entirely and resulted in the project having two years of construction and then a pause. Then the government had to re-contract the whole situation. The result was way more money than originally proposed being spent on the project.
This is not to say that a public-private partnership is necessarily wrong. I am not trying to say that. This is to suggest there is no clear evidence that public-private partnerships will definitively produce a project more cheaply and more quickly than a public entity would. There are examples all over the place on both sides.