In Manchester the actual city centre itself is not gridlocked, its the surrounding roads - M60 at Stockport, M60 / M62 Leeds, M60 Bolton junction, Princess Parkway, Mancunican Way, A57, A54, M56 J7 etc.
A lot of it is school travel as well - the traffic improves massively when it's half term or summer holidays. A CC won't alleviate these roads on the outskirts so what's the point other than being another cost to get to work?
Infrastructure is needed to get park and ride / park and walk functionality in place. Manchester itself needs a massive bypass so traffic heading to Leeds stays off the M60 as that's the only way if you're heading towards Leeds from the west of Manchester. Controversial, but better to have moving traffic than it standing still and polluting the roads people live next to.
A fast train between Liverpool, Manchester, Leeds, York and Newcastle would be fantastic. You could create big park and ride stops in between to encourage people to drive to those and do the rest by train.
What's clear to me is that schemes to price people completely from driving don't work. Not everyone wants to or can cycle 8 miles a day. We need to build solutions for the way people want to commute which will involve the car. Let's aim for part car / part public transport and some issues will be resolved.
Also - cultural shift to let more office workers WFH (where possible) which could take hundreds if not thousands of commuters off the roads daily.