I encourage you to look here: https://www.royal.gov.uk/LatestNewsandDiary/CourtCircular/To... where you can see exactly what they all get up to.
Just like the Athenians found with democracy - where they chose at random their executive each year. The people who run for powerful offices are usually the exact sort who you don't want in charge. Luck is a very good equaliser.
So for constitutional monarchies, the question would be: if they were ever to turn into a republic, who would be their head of state? and how would that be any better than simply staying with the monarch?
[edit: grammar]
No "common" person born in Britain, Canada, Australia etc. today can ever become the head of state of those countries, no matter how smart, qualified, or hard-working they are. That seem benign to you?
Sure, its fundamentally also a symbol of inequality and it is the ultimate anachronism today, but the role it plays in daily life is so low that I don't think it even matters. Do I feel strange being reigned over by an elderly German lady from across the ocean? Perhaps. But the alternative is radical and unpredictable change, and that I am not so excited about. I like the idea that ultimate power in the country is invested in someone who is fundamentally disinterested in politics. The Crown has no political side, only what is best for the country. I understand why people are so uncomfortable with the notion of the monarchy, but I believe it is in the best interest of our country to maintain it.
I would guess that your compatriots from Quebec or the First Nations might feel differently.
EDIT: yes she can choose to abdicate, but not exactly as easily and free from consequence as anyone else could quit their job.
I'm pretty sure she can abdicate if she feels like it.
Fun to think about!
Let's say we decide to abolish the monarchy purely on humanitarian grounds (i.e. it's tantamount to slavery, however gilded the shackles might be, so let's rescue them). Do they get a say? What if they decline to be saved? What does that say about their motivations - maybe they like the rewards of their servitude a little too much? What if secretly they crave power? Does the very act of refusing to be freed imply that they must be freed? What about the converse - if they accept release, then perhaps we'd consider them to be truly moral and therefore ideally suited to the role; mightn't we then decide not to abolish the job after all?
My personal view is that the monarchy is a net benefit to the country, both economically and politically. I believe that republicanism* in Britain is mostly the product of reflexive prejudice and not deep, open minded consideration. I'd be interested to hear what the proposed benefits of abolishing the monarchy are, beyond a trivial cash saving which one might argue is invalid in the first place.
*(meaning abolition of the monarchy, not the Irish variety)
It's quite simple really: in a republic a president is the head of state (as opposed to a monarch), and is elected (as opposed to a monarch) to serve for a defined period of time (as opposed to a monarch). A president is chosen based on merit, not on birth right. It's really about fairness and accountability.
The British monarch acts as a very gentle but constant steadying hand on the tiller of government. The queen meets in private with the prime minister weekly and holds regular meetings with a group of senior politicians (the Privy Council) as well as performing the functions described in the OP. These things help to inculcate British political life with a sense of permanence and continuity, and of duty and service to the country - the latter two being especially well exemplified by the monarchy. The day to day scrapping at PMQs doesn't reflect this on the surface, but read any retired PM's memoirs for some color on how important that aspect of her role really is. This would utterly disappear with an elected head of state, leaving British politics to collapse (some would add 'further') into the kind of rampant short-termism which afflicts most Western republics.
True, she's not elected. True, perhaps, that her position isn't 'fair' in some arbitrarily egalitarian sense, but that cuts both ways: she can't escape her position. The steady state which Britain has reached over the last thousand years of experimentation is proving remarkably resilient, given that supposed lack of fairness and accountability at the very top.
Perhaps it's just that theoretical concerns of absolute fairness and accountability don't really matter when you have a system which works well and allows its people full participatory rights in its government.
That's what Charles I thought......