@JustSayNonce so there's a few things going on here. Firstly, there is a fundamentally important distinction between appealing the PCN, which is a legal question, and challenging the merits of the scheme, which is a political issue. The scheme has been put in place by the elected councillors who run Sheffield City Council, and as far as I can see at the 2023 Sheffield City Council election (https://en.wikipedia.org/wiki/2023_Sheffield_City_Council_election) the vast majority of people didn't bother to go and vote, so they're obviously not too fussed about how the city is run.
You can of course engage in campaigning, start a petition and so on, but the only real recourse is to vote in different councillors at the next election. But even if you get the scheme scrapped, that doesn't affect your PCN, as the PCN is based on the law in force at the time.
Neither the tribunal nor any other legal process can deal with political matters, who the voters want to elect is a matter for the voters of Sheffield alone (we have had some forum members run for their local council themselves in the past, depending on how strongly you feel about this you could have a go yourself).
Putting all political issues to one side, we need to deal with the PCN and can only use the law to do so, and like everyone you have to take the law as you find it (disagreeing with the law is neither here nor there, if you think the CAZ should be scrapped, is a scam or is without merit, I can only repeat that that's something to bring up at the next local elections).
Saying that you'll pay the PCN on condition that the council prove X Y or Z is nonsense, the council doesn't have to prove anything at all to you (and bits of your opening post that mention contracts, corporations and the like have some alarming Freemen on the Land connotations, see https://www.ftla.uk/the-flame-pit/fotl-and-other-misguided-folk/ and https://en.wikipedia.org/wiki/Freeman_on_the_land_movement )
So putting all that pseudo-law in the bin, here's a draft for you to send:
Dear Sheffield City Council,
I challenge liability for this PCN on the ground that the penalty demanded on your notice exceeds the amount due in the circumstances of the case. The penalty charge notice is only permitted to demand payment of the penalty charge, there is no legal mechanism that allows the council to demand the toll charge in addition to the penalty charge on the PCN.
This is akin to a PCN for not paying in a pay and display bay: you can issue a PCN for £70 but the penalty charge notice cannot demand the unpaid pay and display fee on top of that. On this point, I refer you to the decision in Luke Moran v Secretary of State for Transport (IA01249-1803, 13 June 2018) available from LINK.
In this case the PCN may only demand £120, discounted to £60 for the first 14 days. The regulations do not allow the PCN to demand the additional £10 you are seeking, the amount demanded therefore exceeds the amount due by law.
It follows that the penalty charge must be cancelled.
Yours faithfully,
I will PM you a link to put in the representation (one for each PCN), it will redirect to here (https://drive.google.com/uc?id=196wgUERzYDuZjwM9IEFYQwN0jD_jlcDD) but if you give them the link I'll PM you, we can use the click count to confirm whether they've looked at it or not (obviously do not click on that link yourself as we want the click count to remain at zero). If they don't click on it, we can then prove they've failed to consider all of the evidence. If they say in the rejection that they've considered all the evidence, we've got them for lying as well.
When making representations, do it online and get a screenshot of the confirmation screen.