This is an interesting example.
Having an awareness that some people disagree with the concept of monarchy really shouldn't "ruin" anyone's day. It's not necessary to maintain a fiction of total unity in order to celebrate the things you want to celebrate. Banning republican protests on days of monarchist celebration is totalitarian and evil. Whatever your views on any topic, it should be possible to entertain the notion that some people disagree with you without that being a source of grief/upset/spoiling your joy.
Indeed it's really important that it is possible to protest in this way especially on days when the opposite view is being celebrated. If 10,000,000 people are joyfully celebrating the king and 50,000 people take part in a republican protest, you can be comfortably sure that republicanism is a minority view - the freedom of speech means that the opposing view can't hide the (relatively low) level of support it has. If all such protests are banned and gradually over the years the number who come out to celebrate dwindle from 10,000,000 to 5,000,000 to 1,000,000 but it's illegal to have a counter-event so who knows if that would be growing or whether it's just apathy, and then you are actively fuelling the flames of those who want a change - those who are voiceless, oppressed and silenced would always find a way to spread that discontent and you would actually be hastening the onset of the revolution you are overtly against.
Charles is King by the consent of the people. The majority of people are pro-monarchy. If that dramatically changed and 95% of people were republicans and only 5% wanted a monarchy then it would be right and proper to change, and I am sure Charles himself would agree with that. If popular views about republicanism do grow then peaceful protest is a legitimate outlet for freedom of speech to communicate that, but the size of the protest may equally inspire a revival of pro-monarchy sentiment too. Freedom of speech doesn't make either side more or less "correct". You need freedom of speech in order to highlight the falsehoods in your opposition's argument. Obviously no decision would be made on the basis of suze of protest, that's not how democracy works, but if its clear that public opinion is drifting that way it would be appropriate to start campaigning for a referendum on the topic - I would expect that the terms of any such referendum should be along the lines that the question would only be considered settled if the majority was at least 67% to 33% and that any majority less than that should trigger a 5 year process of reforming, improving and making the status quo more acceptable befire running the referendum again - we should never again have a referendum where a majority of a percent or two can impose an enormous change.
Speaking of referendums, would Brexiteers have counted as extremists if this definition had been active in 2015?