WordPress seems like an overkill with most websites I come across that use it.
I’d agree but you are also negating the fact that WordPress from consumer perspective offers a lot of options. From theme selections, to plugins, to availability of developers to hire, just overall resources make it the “go to” platform. At the end of the day it solves problems for many people.
Is there any real downside to not having a database for a smaller site?
As @joe_temp said, you either need one or you don’t. There’s a small overlap (as you mentioned, very simple content management) where you might make either choice.
The real advantage for Kirby isn’t that it uses flat files instead of a database, it’s that it is largely self-configuring and doesn’t require a web host that supports MySQL. It’s convenience. (Not a bad thing.)