There's a reason William and Harry were perceived better than a lot of the core nucleus of the Royals (i.e. not the second and third cousins and so on). Some of it was because of Diana, of course, but they actively did things. Up until the mess with the media smearing Markle, Harry did all sorts (and arguably also won points by leaving the Family to defend his wife and own family). William's fallen more in line with the throne since his marriage, and we'll likely see people sour on him in the same way as his father (perhaps not as badly, but who knows).
There are merits to the tourism aspect of Royal holdings, and the Royal Family that cannot be ignored, but it's not like classical republicans or even outright anti-monarchists don't have valid points either. They're a holdover of an absolutist monarchy, they have safety nets our working poor and homeless can only dream of, and their privilege cannot be overstated.
It's a bit of a mess, culturally, at this stage. I have no problem with their problems being aired. I wouldn't be upset if laws were changed in light of this. But they still command popular support, because of how they tie into the (often mythical) belief of Britain, of England's culture, and all that jazz. So MPs are hamstrung on that front too. Corbyn was famously republican, and that uh . . . didn't do him well (I mean, honestly, nothing he did was reported on fairly, but that was one of the highlighted things).
It'd be nice if we could click our fingers and undo one of the largest relics of the Empire and previous English history. But we can't do that, so outside of your sensible suggestion (which is also, in a way, PR - not putting it down, but that's how it will be used) . . . what do we do?