I don't care how much "real" power they have. If you actually believe in a meritocracy how can you defend someone getting a job because of their lineage, attending real political events and ceremonies and influencing policies that way? And let's not forget that in many cases, they ACTUALLY have institutional powers they just don't use because they don't have to, the institutions serve their interests all the same.
Sorry for the pointless rant. It just happens way too often.
Its precisely because the monarchies and their wealth were never fully destroyed in these countries.
These royals have a lot of power in their local economies because in a lot of cases they still own a ton of estates and land (not to mention investments, connections and as you said institutions).
It is in the interest of the local bourgeois to at least support the continued existence of the royal family as that gives them potential access to these resources.
The royals also create a shitton of propaganda, so some people see them as a needed part of the countries culture or some historical artifact in need of maintaining. A "living monument" or something like that.
And the Pax Americana claimed that Communist countries made Communist leader into cult worship in their double standard...