Personally speaking I think a monarchy brings with it a sense of stability and continuity, so I am not keen to see it removed wholesale. I am also sympathetic to the difficulties of growing up in the public eye because of who one's parents might be, and I am not alone here, nor was I alone in thinking this as an (albeit lowly) celebrity news journalist. However I am keen to see the royal family's modernisation project become considerably more radical. I see this as including;
Paying the same tax as everyone else.
Using state schools and NHS healthcare more frequently.
Using public transport reasonably routinely.
Holding proper jobs more frequently.
Reduced civil list income, leading in turn to reduced media attention and reduced security requirements (the latter being invoked far too frequently in too many situations, like the mobile phone blocking).
And finally
Donating the proceeds of the Duchy of Cornwall to, well, Cornwall.
Indeed I would apply these things to all British residents. We need to mingle more to have a shared sense of values and citizenship. To hammer the royal family for taking paid jobs/using the NHS/sending their children to state schools is unhelpful, as if we believe in equality, and fairness, they need to have the same rights and responsibilities as the rest of us. In this way they can set an example to those members of society that seek a more polarised country with the many serving the few and privilege heavily protected by a veritable fortress of private connections.
That is a long winded way of saying 'noblesse oblige'. And a 21st century monarchy is, in my opinion, do-able, but not in the direction Clarence House is pushing things. Charles is frankly almost Edwardian in his attitudes and this needs to be dealt with otherwise he is likely to be our final monarch.