I cannot follow your logic.
Making a game public will not make people notice it, nor the other way round. Even if it is finished. Itch does not differentiate between finished and unfinished games, unless a user specifically uses that option in browse. There is few places to get noticed for being new. Just have a look at browse and there recent. Look at the games, if they claim to be finished (release status). See a pattern.
I do not want to discourage you, but truth be told, no matter what you do, how good or ungood or finished or unfinished your game is, it probably will not get very noticed either way. There are soo many games on itch. And I did mention that in standard search and browse and tags, feeds, there usually is no difference at all, if a game is prototype, released, or in development, did I?
So the prudent way to do it, is to publish the first draft of the game, as soon as something playable is done. Maybe even start a devlog. Growing an audience if possible. It might not work. But it can't work, if you do not publish. People will not magically flock to your game, because it is finished. You have to have quite a lot of offsite marketing to pull that one off.