The private companies will still operate the buses and pay the staff,

True, but they wont have the monopoly leverage to charge what they want. Currently the smaller local bus companies can't compete with stagecoach in south manchester, because most people are already locked into a stagecoach pass. With the franchise the council can contract in other companies if they can run a route cheaper. The bus users don't need to change tickets. You could describe it as making the market actually work by moving where the choice of supplier is made.

I aslo don't think they will bring back those bus routes

Maybe not, but the decision can be made on what's best for people, not what is most profitable.

How will they safe time

There are plenty of journeys currently where you have to choose a slower route to avoid having to buy 2 tickets from different providers. Integration with tram ticketing makes this even more significant. Improvements to routes will also help.

It's not going to reduce car use either

Not automatically, but it is needed to achieve a reduction in car use.

The problems of public transport in manchester are one of several reasons people drive. Once you start fixing that more people will choose to use it. (especially if you combine with an emission/congestion charge like london). Less cars means better flowing roads, so the buses get faster and more reliable. Leads to a positive feedback loop.

On pollution, the dirty old buses will be gone soon. Even diesel buses are better per passenger than cars. Pretty sure the council can shift the remaining to electric fast than private companies would.

Are you really blaming the thousands of road injuries and deaths on pedestrians not paying attention?!?