It's much more complicated, you don't really have control of when your game will be reviewed.
Sometimes it seems to be quite fast and other times it can take weeks, surely it will depend on how many people are uploading games and the activity of the scammers. So the button you propose would not be of much use in practice.
The good thing is that Itch does not block the page or the download of your game, instead of a button, it is better that you create your own web page that serves as propaganda and that it links to your page within Itch.
By the way, people will search for your game, but most of them will do it in a search engine like google, very few people will search for Itch first and then, within Itch, they will search for your game.
With Steam it is different, because it is mandatory to use the steam application and many of the searches happen from there.