The Queen isn’t the issue, or even really the royal family, although when we get that smug homeopathic tree-bothering-Diana-betraying nincompoop on the throne, I suspect a lot of residual affection for the royal family will disappear.
The issue is what it says about a society, and the way it is structured, the triumph of inherited privilege over merit, the assumptions about social mobility etc etc. The Monarchy by being at the symbolic top of this perhaps acts as one anchor which prevents this changing, and stops people saying… “hey,,, WTF ? “
But we English (I’d exempt the Celtic nations from this) are inert and lazy and reluctant to change, which has saved us (along with the English Channel) from lots of painful turmoil since 1688, and I cannot see the monarchy disappearing anytime soon.