Political correctness fundamentally is at least by default it means treating people, races, cultures, etc respectfully and refraining from using personal attacks, racism, harassments etc in interactions. It does not mean that someone's views or feelings are immune to being challenged or criticized, just that if someone is going to levy criticism or challenge that they do so in a means that is ultimately respectful of the other person or group. Unfortunately in recent times with upswing of identity politics this has gone from 'treat people and groups with respect' to 'anyone who challenges my feelings, opinions, or ideas is the worst common denominator of my opposition.' This is something that in recent years both sides have been guilty of. Cancel Culture has always been a part of political correctness which is why it's becoming problematic given that the needle has been moving towards silencing any opposition as opposed to extremism and can certainly be abused on those grounds alone.
The only group that can silence people is the government, and I'm not aware of that happening right now (the last administration was starting to have a go with the 1776 project).
Outside of that why should people or businesses not be able to choose who they associate with, do business with, etc?
Let's not forget that a lot of this claimed 'Cancel Culture' is nothing of the sort, take this example...
...worn while giving a speech on the House floor which was broadcast live on C-SPAN and across numerous news networks, it's quite frankly amazing how many people are complaining that 'Cancel Culture' is being used to shut their voices down, and doing so live on national and international news network shows! You're not exactly being censored or cancelled if you are being broadcast internationally or nationally!
I don't buy products from certain companies and brands based on the political stance (or other social stance) they take, do you propose forcing me to do so? Now extend that to businesses, why should a retailer be forced to stock products from a brand that they, as a business, do not want to be associated with? Is a vegetarian healthfood shop engaging in 'Cancel Culture' if they refuse to stock meat products? I would be very surprised if you answered yes to these questions.
That same argument extends regardless of the size of a business, up to and including Facebook, Twitter, etc. now you can argue that these businesses are too big and represent a monopoly, but that's quite a different argument. However none of them is required to allow membership to anyone and all of them are free to remove membership from any group, body or individual that they wish to (and they don't actually have to give a reason). The same extends to TV channels, News Stations, Newspapers, book publishers, etc. Unless they are government-owned (in part or in full) then they are not engaged in 'Cancel Culture'.
People, organisations, individually are perfectly free to say what they hell they like, and we, as a society are perfectly free to show them the damn door if we don't agree. Freedom of speech neither guarantees you an audience nor provides you with freedom of consequences. A person is free to be an utter wingnut, racist, conspiracy-theory loving loon, and free to expose those views, that doesn't protect them from being fired, denied service, and avoided by people that used to associate with them. To argue otherwise is to try and provide rights the loon doesn't have at the expense of the rights the others do have.
Pretty much every claimed example of 'Cancel Culture' I have seen (ever) actually boils down to people having to simply face the consequences of exercising free speech that others don't agree with.
You want an example of real Cancel Culture, take
Section 28 of the Local Government Act 1988, which was thankfully fully repealed by 2003. Which, and I quote...
that a
local authority "shall not intentionally promote homosexuality or publish material with the intention of promoting homosexuality" or "promote the teaching in any
maintained school of the acceptability of homosexuality as a pretended family relationship"
...Twitter banning Trump is not Cancel Culture, nor are store not stocking the mad Pillow blokes products (to cite two recently claimed examples), however, a Government making the discussion of an entire sexual orientation illegal in schools, colleges and any other government-provided service is. The first two (Trump and Pillow) are simply a result of the consequence of actions, the last is a denial of rights and an abuse of power.