Mine does the same thing. But if you use Firefox or Brave or some other browser, you can download the beta profile from Apple. You need to then send it via email (using iOS Mail App) to yourself. Open Mail and click on the Beta Profile then you can install it on your iPhone.
This is not working for me. I already had the Developer beta for iOS/iPadOS16 on an old iPad and old iPhone, and they've been stable enough that I was willing to switch over to the Public Beta on those two devices and my current two devices.
However, from https://beta.apple.com/sp/betaprogram/welcome in Safari after I click Sign In, I apparently sign in successfully as there's a Sign Out link in the top right, but the Sign In link is still at the bottom of the page. There is no profile to download.
I had to go to https://betaprofiles.com and select Public Profiles to download the beta profile. Then I followed the cues.
FWIW, deleting the Developer Profile from my iPhoneX, then installing the Public Profile, then rebooting and checking for updates resulted in no update available (because the two builds are the same at the moment?). The beta was waiting for me on my iPhone 13 Pro Max. I guess future beta downloads will be a few days delayed now that I'm on the public beta.