A lot of people have a tendency to want to show loyalty to their country by showing subservience to some rich arrogant asshole that surrounds themselves with gold. See the how a significant percentage of the US population think of Trump for example.

If there is no King the subservient portion of the population will create one. It's better to have a King that doesn't have any actual power so that won't happen. The subservient part of the population don't really care that their King has no political power, they just want to bow to someone to prove their loyalty to the country and see some pomp and pageantry. For a lot of people the concept of a country is too abstract to understand so they need some person to do all kinds the ceremonial stuff so they can express their loyalty to the country by showing loyalty to that person.

Similar to how having a separation between government and religion, it's a separation between government and all the ceremonial pomp and pageantry stuff.

Sure, I wouldn't directly care if there was monarchy was eliminated, but a lot of other people would. And those people would start voting in some wealthy asshole to rule over us like a king. And that's something I definitely don't want. So just give the subservient types someone they can bow to so it doesn't impact the rest of us.