Then why not leave things as they are and, as you say, carry on as usual.
Unless, of course, some demonstrable advantage can be shown by instituting the changes you suggest.
There is one principled objection, and several practical ones. I'll draw examples from the Dutch monarchy, but I think the Dutch and British/Aussie/Canadian/... monarchy are close enough in how they work that those objections equally apply to the British case.
The principled point is that the function of monarch is not open to anyone, but only to the descendants of Sophia of Hanover resp. king William I of Orange. Every function, whether public or private, is open to anyone, but only one specific family can become monarch. That is fundamentally undemocratic. The historic rationale behind it is that nobility, and royalty in particular, are somehow "special" people. Nobody believes anymore in that pretense, much less the royals themselves, as they now routinely marry commoners and not other royals/nobles. Our king William II, mid 19th Century, married the daughter of the Czar; our current king married the daughter of an Argentinian mass murderer. Hmm, looked that way not much has changed.

(The additional argument brought by Orangists is that our royals are descended from William the Silent, which is nonsense because the current ones are descended from his brother).
My first practical objection is that a monarchy means a royal family, with a lot of good-for-nothing parasites hanging around it who live on the public purse: kids, grandkids, siblings, cousins, they're all part of this royal family and must be maintained. On the flipside, I concede, they may also serve as auxiliary ribbon-cutters.
My second practical objection entails all those minor annoyances and crises, caused by the abuse of rights and privileges by not only the monarch, but the whole wider royal family, people who were born with a golden spoon to begin with. The Dutch constitution plainly says "the king is inviolable, the ministers are responsible", so the monarch gets away with what he does, and so does his family. In the last ten years, we've had the queen, more probably her father,
ordering the intelligence service to spy on her niece and her husband - an affair we haven't seen yet the end of; a
controversial holiday home of the crown prince (now king) in Mozambique; and the
queen's sister administering her Guernsey tax evasion scheme from the queen's royal palace. In the British context, I could think of Charles' continuous letter writing to ministers.
My third practical objection is in line with that: the major constitutional crises. In the Netherlands, we came close to the abolishment of monarchy in the mid-1970s, when it turned out that prince Bernhard had received bribes from Lockheed (and Northrop). In the cabinet, Catholic vice-PM and Justice Minister Van Agt argued, with a large minority in the cabinet, that Bernhard should be prosecuted as any common criminal, but Labour PM Den Uyl shied away from that. In case of a prosecution, queen Juliana would have abdicated and princess Beatrix would have refused the throne, so a near-certain end of the monarchy would have resulted. The choice is to let a high-profile criminal go free, or to have to rewrite the constitution on the spot.
A German-style presidency does not suffer these problems. A president does not bring with him a family that is owed reverence too. And a president with a scandal can resign, or be deposed, as the recent case in Germany shows: Christian Wulff was accused of corruption, he resigned, new elections were held and Germany had a new president.
Mind you, I think your suggestion, at least in some measure, would fall under MG1962's ambit of "loosening a screw here, tighten a bolt there". My main objection is against those who would prefer to throw out the whole machine and install something completely new without really understanding how either the old one worked or how the new (as yet undefined) one would offer any benefit.
The number of people who argue against me that we need to end the ability of the Queen to arbitrate matters of high import is quite depressing sometimes, given that the people making such arguments obviously don't have a bloody clue how our government actually works.
It's certainly not my intention to throw out constitutional history. I'm aware that the actual interaction between parliament, ministers and monarch is something that is grown over the years and decades, and is largely a matter of custom and tradition rather than black-letter law. In fact, while no-one seriously argues in the Netherlands about abolishment of the monarchy (damn you, Maxima), we had a serious constitutional novelty in 2012. Custom had it, in the Netherlands, that after elections, all party leaders would visit the queen and advise her on which coalition to form. The queen, then, would appoint an "informateur" who would probe the viability of one (or more) coalitions. This is, IMHO, the greatest actual power the Dutch monarch wielded, but it was nowhere written down. After the 2012 elections, parliament decided to take matters into their own hands, and appointed an informateur themselves.
To be absolutely clear: my only point is that the hereditary principle of the function of "head of state" is archaic and should be abolished.