Democratic nation doesn't topple when one politician does something unethical or illegal. And firing the president and all other politicians is part of the systems functions when they do something wrong. So, unless someone mabages to change the coumtry from democracy to something else the country has not failed.
Meanwhile in a monarchy, accidental deaths or even a dispute over who should inherit the throne can destroy the country or at the very least change the ruling family.
And that's just few of the many reasons why democracy is the least bad form of empowering the rulers of a nation.
An impeachment over a political scandal can cause a political crisis that can affect the countries stability. So a democracy has an equal probability of becoming unstable.
If a monarch dies, there is a stable system of succession, and the succession laws can prevent succession disputes. The best succession system which was done in most European Kingdoms was male preference primogeniture, because it makes a royal hose last for a long time, but has an efficient system, in case a monarch has only daughters.
Do you even read what you are typing before pressing "post?"
If you have an insane king that the people overthrow, or there's a succession war between different heirs or anything like that, it basically results in the original line ending and a new ruler taking the throne.
Meanwhile, politicians have been arrested for crimes or lost their seats due to incapability to do their duties but this has been built into the system and it does not deligitimize the democratic leadership. I'd even argue that holding your leaders to be under the same laws as everyone else is proof of strong democratic foundation.