But is there an alternative?
Now, I'm not a monarchist, but I think that within the setting of GoT, there's actually some merit to keeping the Targaryen dynasty around, since a nice democratic republic doesn't seem to be a realistic option.

The problem is... what exactly is holding the Seven Kingdoms together at this point? I mean, we've got powerful elements of the North, the Iron Islands, and probably others who are probably just itching for a chance to declare independence. Under the first Targaryens, it was held together by dragon fire. That's gone (for the time being). All that's left is... law, and tradition. Take that away, and the only thing holding the realm together (or more likely failing to hold it together, since no one kingdom is powerful enough to dominate all the others without dragons) is naked force. IIRC, that was basically Renly's argument, when he made his bid for the throne. That the only real legitimacy left is "I've got an army". Which means an endless cycle of war as every ambitious person who can get some armed men behind them makes a bid for the Iron Throne, at least until someone is strong enough (like Danaerys) to hold it all together by force.
So, what if, rather than putting Robert on the throne, they had instead simply removed Aerys (and falsely-accused-of-rape-Raegar, I suppose) on the grounds that they have violated their duties as King/prince (IIRC, under feudal systems, lords generally had responsibilities to their vassels, as well as vice-versa), cut off their heads or given them a choice between the headsman and the Wall, and put one of the other Targaryens on the throne? Could this have worked? Would their be a legal or cultural basis for such a move in the setting?
Of course, there is the little problem that the next in line, if I'm remembering my Targaryen family tree correctly, is freaking Viserys, perhaps the one character who might actually be more of a vile, useless nut job than Joffery. But there is the nature vs. nurture question- would Viserys have turned out so badly if he hadn't grown up an exiled orphan? In any case, in this hypothetical, a precedent would have been established that while the dynasty endures, a king can be removed if he fails in his duties to his subjects.
Perhaps this could even be a first step towards some kind of constitutional monarchy or democracy- a understanding that while the lords are loyal to the crown, their are some lines that the king cannot cross without the Lords Paramount removing him.
So, would it have been possible to overthrow Aerys and Raegar, and put Viserys on the throne, with the queen or one of the victorious rebel lords acting as his regent until he came of age, thus preserving the legitimacy of the monarchy? Would the other kingdoms have accepted this? Would Robert's ego have accepted it (yes, he grew to hate the throne, but that was after years of unhappy marriage and disillusionment with the responsibilities of being king)?
Or am I completely off-base here?