As far as I know there isn't a toggle like that for a normal public page right now. Free downloads on itch don't need an account and don't create ownership, so anyone with the link can grab the files. The closest thing today is download keys: when you generate them, you can require the key to be claimed before the files can be downloaded, and claiming means logging in or making an account (/updates/download-keys-can-now-be-claimed).
That works for a small group like testers or supporters, but it's a lot of manual work for a mod with lots of downloads, so an optional "registered users only" setting does sound useful. Nice to see someone this fond of the site.