Is there a single government service or dept which has been privatised and it’s turned out cheaper in the long run for tax payers? I can’t think of a single example.

Nothing comes to mind. Because even if the original goal of privatization is to bring costs down (which I’m sure happens in many cases) the overall LONGTERM goal of a private business is to return money to shareholders. So in the very short term it is totally plausible for costs to go down (capture low hanging fruit efficiencies), but long term this will never be the case, because the system is designed to extract as much wealth as possible - once you’ve captured cost/expense efficiencies, the next step is to capture pricing efficiencies.

This I believe will be a very bad outcome for bottom 90% of Americans and a very good outcome for top 10%

It’s the same myth that universal healthcare would cost people more than private insurance. Insurance is there to make money. Gov programs don’t need to be profitable - just sustainable.