So, I think it's self-evidently true that vocal minorities can make their views felt strongly. However, I think there's more to unpack here than that, and there's a large difference - as you point out - between a majority acquiescing passively and one being forced to do so by a state apparatus. I think it's also very frequently true that political debates are led by vocal minorities on both sides of the argument, and I'm not sure this is a problem - indeed I suspect it's an inevitability.
To take your examples: broadly speaking, food labelling is an issue of majority acquiescence. There's no obvious preference not to do X, some people want X, it makes sense for markets to provide X. That's 100% fine and dandy.
Politics and political correctness are a much more difficult issue, for a number of reasons:
> Firstly, whose preferences count when we take into consideration what the "driving minority" is? Those of a party's MPs, or its members, or its voters, or the wider public? It's pretty clear that Corbyn broadly represents a majority view amongst Labour's vast membership on most issues (EU being the obvious exception). His faction are a dominating minority among MPs, but are able to hold a plurality of the electorate and a majority of the party membership in situations where other options are offered.
> Regarding political correctness; "political correctness" issues I don't think are a very analytically useful category to start with, since depending on who you ask things seem to vary between political correctness meaning "argued overreaction to a linguistic gaffe" and "saying transgender people should have human rights". I also really don't think this classifies as a true "intolerant minority" issue, because I think the group railing against "political correctness" essentially forms a second intolerant conservative-leaning minority, with the preferences of the public being more apathetic on either side. Or to put it another way, a successful intolerant minority isn't noticeable because they just sail through and win (for example peanut allergy sufferers in the article). Nor do I think you can classify the conservative vocal minority as a backlash - rather, in many cases they represent the current vocal minority position. To go back to trans rights: the idea that transgender women should be treated as women is currently not a position effectively imposed by the vocal liberal minority who believe in it. The reason it isn't is that there is a similarly sized vocal conservative minority trying to rally sections of the public in favour of permitting discrimination. The fact that any significant section of the public even has much of an opinion on this sort of thing is solely because it's where a political battleground has been pitched: were it not for there being two vocal minorities, the majority of the public would just acquiesce to the views of one side or the other.
>The above - the public essentially being a football between two groups of people actually engaged on an issue - isn't something we should treat as a problem. It's how literally all politics works, always. The things we should be concerned about on a systemic level aren't the levels of intransigence of viewpoints in the two groups; it can be entirely reasonable to be intransigent and uncompromising in your viewpoint, depending on the situation. If your opponent's viewpoint is simply morally abhorrent, or factually baseless, you can't meet them in the middle and it's unfair to ask people to try. This is one of the classic tactics of the far-right, to claim that we need to "have the debate" on things like "are white people superior" - it's unreasonable to ask anyone to come to a middle ground on an issue where the answer is "no, sod off". The systemic issues we should be more concerned about, I think, are whether the systems for resolving these disputes are adequate in terms of informing, and accurately representing the views of, the majority, and in ensuring that vocal minorities can't win "victory by conquest" aka winning the argument and thereby disenfranchising their opposing vocal minorities.
> In terms of how much we should tolerate intolerance: broadly speaking, I believe in freedom of speech with a few limitations, but also very firmly believe that freedom of speech is something that only operates vis-a-vis the state. I don't think we should go around arresting hard-right (or stalinist-left, but those are rarer) politicians, but I don't think we need to give them any more voice than they need, and I think we need to be aware that there are certain viewpoints that are fundamentally opposed to democratic values and that it's reasonable to ask that the services we use don't give them funding and oxygen. It's important to recognise that groups like Stop Funding Hate are very dependent on how large a percentage of the population they can mobilise, which I think is true of a lot of boycott-style pressure groups - this I think is a check on their abilities. If the Mail readership are desperate for the Mail to keep being published, they can crowdfund it tbh. This is also why I don't think student unions "no platforming" fascists is in any way bad or a problem - fascists can go hold meetings somewhere else, free speech means the government won't arrest them for it, it doesn't mean universities are obliged to give their views the time of day.