If a future British monarch did not have approval of the majority of the public, would the institution of monarchy be negatively impacted?

To a certain extent yes but, Charles and his equine wife, will certainly not be popular however, an unpopular monarch does not necessarily mean that the monarchy itself will be unpopular. We will just have to endure the future 'Charliean' era and look beyond it, to the future reign of William.

Not at all, the monarchy answers to no one, it would take an act of parliament to remove a monarch.

I suggest you don't get your hopes up re the UK abolishing it's Monarchy any time soon. In general, we don't want to have to elect a politician to take over the duties of a Monarch. Regardless of who that Monarch might be.

The monarchy exists because the majority of Britons prefer the constitutional monarchy. If the majority of Britons did not want the monarchy, there would be a referendum leading to a vote, by the people, to end it.

The monarchy is there historically by consent. If enough people wanted a change it could happen, but there are no indications that is on the cards in Britain where the monarchy has never been as popular as it is currently. The lack of any credible alternative which does not involve politicians is also an important factor in its continuing popularity.