Most absolute rulers, be they Monarchs or dictators, will accept no diminution of their power, so that when the populous approach them with such a suggestion they, the absolute ruler, relinquish some or all of their power,they tend to reject the demands of the populous.
If the feelings of the populous are strong enough then the populous will persist with their demands, and typically, the ruler persists in their rejection, and thus tension in the country builds. That tension often reaches a point where it explodes into outright confrontation, (something of the sort can be seen in countries around the world today), and when this happens the fight then becomes all-or-nothing. That is 1 side becomes the outright winner and the other the loser. If the ruler loses then either the ruler gets executed, as was the case in Romania in the latter half of the last century, or perhaps the ruler is given the chance to flee into exile.
The point is, that by the time conflict erupts within a country the opportunities for a peaceful and negotiated transfer of all or part of the rulers power is generally lost and so monarchies tend not to evolve into parliamentary or democratic monarchies. Of course, this is a generalisation and over simplified, but for the purpose of your question I hope it provides something of a an answer.
PS, as a British national I cannot agree with your statement that Monarchs are symbols of national unity and pride. Even a cursory examination of the history of the 'british' royal family (many of whom throughout history could not even speak english) shows them to have been ruthless in their determination to retain power, and careless towards the sufferings and plaints of the people they governed.