Someone from the shop looks at every sticker before it joins the tray, so yours may take a little while to show up. If it fits the shop, everyone gets it.
Untick the platforms you don't own. The AI Catbot then only recommends games you can actually play. Asking for a platform by name (“any Switch gems?”) still works and overrides this.
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Two ways in, and you can use either or both. Pasting your library is the only way that brings in family-shared games, since Steam hides those from everyone but you.
A pasted list carries no playtime, so those games come in as Not Played and you pick which ones you played. A profile URL adds playtime and pre-sorts anything you have played for two hours or more, but it only works if your Steam Game Details are set to Public, and it leaves out family-shared games.