Problem is how accurate are the figures. Or rather how can you say with any confidence that they are accurate. I have never seen anyone produce any, for either side of the debate, which weren't seriously flawed.
The checks and balances wouldn't be improved by effectively electing the monarchy. The powers side of the monarchy is different as the political powers of the monarchy (as in those it effectively possesses) are minimal but those it could in fact have are not. However the reality is if the monarch was to, for instance, refuse to accept certain ministers or pass laws then bar a truly bizarre situation where the country favours the hereditary monarchy over their elected government then it is likely the constitutional balance would be altered. That the government serves the Queen by name and not the people is a semantic. The government is answering to the electorate already, changing them to the people's government isn't going to change how in or out of touch they are.
The problem with the argument it would clarify the leadership of the country is that it is difficult to see how replacing the Monarchy would either a) particularly achieve this b) whether this a real issue. If we accept that the UK would move to a German style system whereby the President is just a perfunctory role and thus isn't truly politically powerful and not as in USA or France a politically powerful figure (which we should do as it is the only realistic change as the other requires too much constitutional overhaul). Then the PM would effectively remain the head of state and people will continue to discuss voting for that person or their rival at a GE. I would suggest that is as much a result of how politics and the general population interact with many not being overly interested in their backbench MPs activities nor do they have much impact upon them.
I also reject the idea that suggesting there are bigger issues is apathetic. Republicanism is a losing issue which few care deeply about and which few can suggest a change in the situation would have a material difference on how government operates on a regular basis. To focus on that and miss out on the West Lothian question, devolution, House of Lords reform etc is a waste of time as the other three issues are much more likely to gain traction is how to lose the constitutional reform argument.
Not sure many can or would bother defending the existence of the royal family but that isn't the point. The point is that they don't need to as it is the status quo which functions fine and there is, as I have said before, little to suggest that altering it would make a particularly big difference. We aren't going to suddenly see an executive which can't just remove civil liberties or pointedly remove benefits from the poor because we have changed how we decide who holds a ceremonial role. It is a heavy duty constitutional reform for little effect. Compared to producing a functioning House of Lords (which is a seriously complex issue in its own right) or clear local governance it really isn't that important to most people's lives.
I don't disagree with being against the Royal Family, I just find it weird that given the other important reforms needed that this riles people up so much. I would be much angrier about people being effectively born into the legislature or getting there via the state religion which discriminates against other religions in an increasingly secular country. Than people who the removal of will not greatly alter the political balance