An apolitical head of state who can serve as an additional check and balance on a crazy head of government and legislature makes sense.
The US system has checks and balances by having branches (an allegedly independent judicial branch, an executive and a legislature) that are designed to block the crazy elements in each other branch, but if crazies get elected to the legislature and the executive (and they are from time to time), then they can also pack the judicial branch with crazies, and the country is then doomed. Laws are only as good as the people enforcing them.
I'd definitely prefer to have someone immune from partisan politics who can at least say NO and slow down the crazies in the rest of the government from turning the country into a tyrannical state.
I am actually skeptical (BrEng "sceptical") about the ability of the Head of State to say NO to the elected government or parliament in modern constitutional monarchies The last significant time that happened was probably in Australia in 1975 when the Governor General dismissed the prime minister when he refused to resign or call a general election after failing to pass the federal budget in the Senate. The end result of that action was to make the Australian Labo(u)r Party embrace republicanism, thus weakening the monarchy, rather than strengthening it.
In more recent times, what we have seen throughout the Commonwealth is an increasing subservience of the Crown to the will of the partisan politicians. In Canada for example, Governor General Michaëlle Jean agreed in 2008 to a request by prime minister Stephen Harper to prorogue parliament in a deliberate attempt to block a vote of no confidence in the government, even though it would be within her discretion to deny the PM's request, as she probably should have done IMHO.
In the UK, on the other hand, we would probably have to go back to William IV in 1834 for the last time a monarch oficially refused to follow the advice of his ministers and for Queen Anne in 1707 for the last time a monarch actually used the legislative veto power. More recently, there was speculation, even among many legal scholars, that Queen Elizabeth II could play an active role in government formation (and thus serve as an impartial, non-partisan "mediator") when the 2010 general election returned a hung parliament. That didn't happen though as the Queen was completely excluded from any coalition negotiations and only played a (symbolic) part in the process after the Conservative/Liberal agreement on a coalition had already been sealed and the Labour prime minister, Gordon Brown, had formally submitted his resignation.
I suppose in some countries like Belgium, or perhaps Denmark and Norway, the King still has some residual political influence, but that is definitely not the case in the UK and the Commonwealth, where any political power the Crown still has in theory is now mostly symbolic. Sweden went one step further in 1975 and decided to abolish the constitutional fiction altogether, stripping the King of any formal role in government formation and in the legislative process. In that sense, one can no longer talk of the monarch as a "check" on the government as far as the Swedish constitution is concerned, even though he retains the constitutional right to be informed about government affairs and to give advice and express his opinions to the cabinet, and, most significantly, has kept his position as the ceremonial Head of State.