Is it fair, and is it private?
The draw uses your browser’s cryptographic random source, the same one websites use to make security keys. Every possible draw where nobody gets themselves or an excluded person is equally likely.
The names stay in your browser. Each person’s match lives inside their own link, after the # sign, and browsers never send that part to a server. We don’t store the draw, we don’t send emails, and nothing about the names reaches our analytics.
One thing to know: whoever creates the links could open them. If the organiser is in the draw too, they should send the links without opening them, which the tool makes easy because the list shows names, never matches.