Is there a way to search for games that are complete? So many developers never say that they're done with a game or mark their page in a way that make it easy to tell that they've moved on to other projects.
This could be something you suggest
It is all there, but this is not enforced or curated and we are talking about mostly amateur developers and small time indie deves here. You will find finished games that have status in development. And released games that get content updates. To make it more complicated, in the hover screenshots, in development for paid games is called early access.
When is a game finished? There is not even an agreed upon version numbering. There are devs that use 1.03 version for unfinished games that are less finished than dev's games that use 0.1.3
One can guess from the description, but searching from the outside is unreliable.
You need to scroll down to Directory on the page, in the footer. https://itch.io/directory
In theory, the status released would denote a game that is "finished" and ready to be deployed to customers. But in practical terms, many devs take this to mean "published" and they "release" even their very first draft to public eyes. Maybe it is a language thing. And others forget to change status from in development to released. And just because something is released does not mean, development is finished. That is not even accurate for AAA games, they often "release" unfinished games that are barely playable.
In my book, you should be able to play a game from start to finish with all major intended features to quailfy for released status. But there is no enforcement for any of those release status terms.