This appears to be a widespread issue occurring today, August 21, rather than a problem specific to your PC or phone.
A rapidly growing number of users are reporting the exact same Phone Link behavior today: the Continue button does nothing, account switching does not work, and in some cases Link to Windows also fails to sign in. Users have already tried Repair, Reset, reinstalling Phone Link, clearing caches, and even testing different Windows profiles or PCs without success. See this other thread: https://learn.microsoft.com/en-us/answers/questions/5982147/phone-link-stuck-on-use-your-mobile-device-from-yo
There are also independent reports from users in different locations experiencing the same failure today. https://www.reddit.com/r/PhoneLink/comments/1vu7wp9/phone_link_not_letting_me_sign_in (note, non-Microsoft source)
This strongly suggests a Microsoft-side Phone Link authentication/service issue. I have not found an official Microsoft acknowledgement yet, so I would avoid reinstalling Windows or making extensive system changes for now. Report the problem through Phone Link > Settings > Feedback & Community or Feedback Hub (Win+F) and wait for Microsoft to resolve or acknowledge the issue.