Create an exchange, share one link, and everyone joins from their own phone. When you draw the names, each person privately reveals the one person they are buying for.
You get two links: a join link to share with the group, and a private organizer link for running the draw. Save your organizer link.
Open the join link the organizer sent you. It takes you straight to the exchange page where you add your name and wishlist.
Your private organizer link is saved on this device. The draw pairs everyone at random, nobody draws themselves, and exclusion pairs picked at join time are honored.
Your wishlist is shown only to the person who draws you, after the draw.