You'll expect me to come in now and say Public Good Private Bad. And I won't- it's a lot more nuanced.
So, to start. There are good public operators (Lothian) and good private operators (Southern Vectis). There are terrible public operators (Warrington) and terrible private operators (Stagecoach Busways). Ownership isn't the defining factor.
So, "innovation". It's hard to be truly innovative with buses, and there's no evidence the private sector are better or worse at it. I remain sceptical that glossy seats and USB charging points tempt anyone out of their car. London buses don't have them and they have ridership other major cities would dream about. And London Buses' livery- all over red- would be derided as nasty anywhere else in the country.
What works is bit of stick- expensive parking, traffic jams- and easily accessible information about the alternative. Best Impressions, love them or loathe them, are GOOD at the information behind the shiny livery. Information is easy to read and easy to obtain. That's what gets people to see the alternative to sitting in a jam fighting for a parking space.
Secondly, you need coherence in a network. Franchising gets talked about in areas where there is no coherence, like in Tyne and Wear. Each company operates in a bubble. If you have to cross into another bubble it becomes staggeringly expensive.
London is the very definition of coherent: I have one card I can use on all buses, all trains* and all tubes. Hell, I can use my bank card and they bill me once at the end of the day/week. It's simple. It works. You don't get that anywhere else, and it shows. (*yes, yes, HS1 and HEx, I know).
But also, there's no coherence with infrastructure. Councils get pressured into building bus priority measures, only to see operators not use them. There are bus gates in South Tyneside that cost a fortune to build, at Go's insistence, but now don't get used because Go changed their network a year after construction. You see the same in West Yorkshire: shiny bus shelters and raised herbs on roads First abandoned with 56 days' notice. You can criticise councils for not investing, but when they do invest but their money gets spaffed up the wall by operators changing routes on a whim, what do you do? Once bitten, twice shy, after all.
For me, the issues come back to how things were privatised. There is a reason London had different rules. Operators can't collaborate as it's a cartel, but integration requires collaboration. And without integration you have nothing.
As with the railways, so much of what is wrong boils down to the stratospherically cack-handed way things were privatised, rather than the actual privatisation. After all, the oldest omnibus companies started out in private hands.