The question then is who carries out the representational roles in the republics? (If the answer is no one, that would create evidence that the restriction or elimination of these roles is not necessarily unacceptable to members of the public.)
I think we are talking about two different levels of representation roles.
First, an i
nstitutional representation role that includes e.g. hosting visiting heads of state, accepting foreign diplomatic credentials, or giving out orders, medals and decorations. That kind of role is fulfilled in monarchies by the king/queen (or equivalent), and in republics by the president.
The second level is patronage of national academies, universities, and diffferent arts, science or sports associations; or involvement with charities, NGOs or miscellaneous social or environment groups, including multilateral international bodies. That is something that members of Royal Families (not only the monarch, but also princes/princesses) do quite often. Furthermore, they also have a close connection with military institutions, including veterans. By contrast, presidents normally don't have that kind of role, with maybe a few exceptions.
Whether that role is missed or not depends on the expectations that a given society has. My impression is that, in European countries that are monarchies, the public has grown accostumed to seeing royals involved with miscellaneous patronages, so, if they suddenly and abruptly withdrew from those roles, they would be missed.
I am all for a slimmed down monarchy. But with all the patronages they have, I’m not sure how they will manage it. It makes sense to me to have Charlotte and Louis as working royals should they so choose when they are much older and really understand what that does and doesn’t mean. I don’t think that Andrew, Edward , and Anne thought they should be treated in the Firm the same way as Charles. But I agree that Charlotte and Louis’ s children should not take on these roles. Honestly, part of it boils down to the number of children they have. What if George has one? Then what?
I have read a number of things that made it clear that Diana was keen to treat the two boys equally - that makes perfect sense for a parent and should be true for family relationships. However, I wonder if Harry never really got that that meant he would not be treated the same way in the Firm? A bit more of a challenge since Charles had only two children.
The criteria for "working royal" from Edward VII (?) to the present seems to have been primarily:
1) The monarch and his/her spouse.
2) The Queen Dowager, when applicable.
3) The monarch's sons, including the heir (and their respective wives).
4) The eldest living sons of the monarch's sons (and respective wives).
Note that the monarch's daughters were not automatically working royals
unless the heir happened to be female (e.g. Princess Elizabeth) and/or the King had no sons and only daughters (e.g. George VI, in which case Princess Margaret for example was also a working royal). Queen Elizabeth II innovated by having Princess Anne as a full-time working royal. Note also that (4) above included for example the current Dukes of Kent and Gloucester, but excluded Prince Michael. Similarly, it would normally have excluded Princess Alexandra, but Queen Elizabeth II drafted her anyway. It also excludes Beatrice, Eugenie and Louise, but not James.
The exclusion of the female lines from the working Family actually automatically implied a slimming down, which was consistent with princesses not getting peerages and not passing down royal titles. Longer ago, the primary fate of a British princess would have actually been to marry a foreign prince from another royal family (or, more recently, a British peer), rather than serving domestically as a working royal.
The introduction of equal primogeniture, under which male lines are no longer preferred over female ones, might force a revision of the criteria above, but that is not a given. If there is an alternative gender neutral slimming down criterion, the natural option is to keep as working royals:
1) The monarch and his/her spouse.
2) The monarch's living parents.
3) The heir and his/her spouse.
4) The monarch's children other than the heir.
5) The heir's children.
Note that (1)-(5) are the persons who are now entitled to royal styles (Majesty or Royal Highness) for example in Spain. Groups 4 and 5 may be extended with wives of the monarch's sons or of the heir's sons although husbands of daughters are normally not included unless the daughter in question happens to be in direct line to the throne. Group 5 can be, on the other hand, further slimmed down by keeping only the heir's eldest child as a working royal until the heir ascends the throne.
Any additional slimming down, e.g. exckuding all of the monarch's children other than the heir, would be too extreme IMHO for a country of the size of the United Kingdom.