Itch, unlike Steam, has a self-publishing model and is designed with a "pay what you want" approach.
This means you can publish your project with almost no restrictions. You can publish games, programs, books, comics, music, etc.
You can publish only for Linux, or only for Android, or an HTML game.
Unless you commit something illegal or try to upload malware or a virus, Itch won't prevent you from publishing your project. However, Itch doesn't index all projects; that is, it doesn't add all published projects to its internal search engine; it has a certain minimum level.
The big problem is that Itch doesn't have the resources to manually review all games. It has an algorithm that, based on certain criteria (which the staff has never revealed), allows it to be indexed or placed in a queue for manual review by a human.
Since Itch doesn't have a large staff, that manual list can take a few hours, days, or even weeks, and I've read about many games that have been left in limbo for months.
So, "published" is something you control. Your page is public, it can be visited, your product can be downloaded/purchased, etc.
"Indexed" means Itch adds your project to its internal search engine if it has previously been accepted by the algorithm, or if not, if it has been manually approved by the staff.
Let me clarify, only projects can be indexed, not developers. If you search by a developer's name, Itch shows you that developer's indexed projects, not their profile.
If the developer does NOT have any indexed projects, then the system will not show you anything.
Why not a wishlist?
The "pay what you want" model is the most used on Itch. In that model, it doesn't make sense to have a wishlist, since the majority of games published and consumed are free or pay what you want (1,200,000) vs. paid (32,000). Instead, Itch only implements a collection system, where each user is free to group the projects that catch their attention in any way they wish. If someone wants, they can create their "wishlist" collection, but Itch doesn't offer that as a ready-made option, because Itch isn't focused on paid products, as Steam or GOG are.