The German and Irish Presidents have no real power, if you want that have a French or US powerful elected President, nor advertise the country with brand recognition as a ceremonial head of state as our monarch does. If you replace the House of Lords with an elected Chamber then you also get US style deadlock, much fewer laws would get passed as the second chamber has a mandate to block the decisions of the Commons. At the moment we have a second chamber with plenty who have been top of their profession.Well, so what, really. The Head of State is meant to work for the country and be a host for those from abroad. If all we want is for the Head if State to be performing monkeys for tourists, then that's great. That's what we have. I think the whole argument is moot anyway as we won't get rid of them. There are far more important things constitutionally wrong with our system, ( much of it because of our culture of deference- the HoL, our obsession with dumb posh people telling us what is best for us) stems from us being a monarchy. They deserve to be criticised precisely because they are not held to account in any other way. The press is sycophantic in the extreme, Parliament rolls over and allows them to change laws to benefit them, the police allow them to do PR at vigils before arresting other people, and allows them to get away with not being investigated or being subpoenad as a witness in major court cases. Monarchists would just bend over and let them get away with everything they want.
As for the so called sycophantic press to the monarchy have you ever read the Guardian or Kevin Maguire in the Mirror, full of Republican propaganda and Murdoch is hardly a monarchist either. Maybe if you went to a republic like Russia or China or Syria you might see what real media propaganda is and opponents don't even get to air their views like in a constitutional monarchy like ours, they get arrested instead