Many of the countries on the list of constitutional monarchies are currently stable. For example, is there a serious argument that there is a problem in the constitutional monarchies of Sweden, Norway, or Denmark? They are three of the most egalitarian and democratic societies of the world.
Changes can have unintended consequences. For example, in most constitutional monarchies, the sovereign can dissolve a government and force a new one to form. As a result, they may not have another law allowing that to happen in emergency circumstances. What happens when such a law is really needed?
Now, if you're asking if a country should add a sovereign where there wasn't one previously, that's harder to argue. Then the stability argument cuts the other way.
Retaining constitutionally limited monarchies also has the effect of making it easier for absolute monarchs to cede some power. If an absolute monarch can see the wealth and influence of the British monarchs, it may be less worried about passing power to others. This encourages them to pass power to moderate dissenters rather than wait until forced by revolutionary extremists. In general, constitutional monarchies are more stable than dictatorships.
Most monarchies started because powerful people preferred that monarch to alternatives. In most countries now, this would be regarded as a step back. But there are some countries that are ruled by local warlords that might benefit as a result of switching from multiple warlords to one monarch. Giving the monarch the benefits incents him or her to control the warlords. Having one law for the entire country encourages commerce.