There are two types of searches. "Search" and tags. Neither searches for content/subject. Try a regular internet search engine for that. They should accept things like
site:itch.io play as a ruler
The search box gives you title matches and partial title matches and maaaaybe some games that have terms in their short description.
The tag box gives you, well, games that have been tagged by the developer with that tag or genre. But since this is free tagging, your guess is as good as mine as to what tags would be used by devs of games you seek. If the game even has proper tagging to begin with.
If you find one, expand from there, look what tags are used. Maybe you find a bubble of games interconnected by specific tags.
I would imagine you can find some with those tags: sandbox, management, simulation, kingdom, medieval, court, king, queen, prince, princess, ...
And of course, in the search box, you can find games that have royal terms in their name. They will use titles like king something or something queen something or something kingdom and so on.