iCloud sign in on the web keeps bouncing me back to the login screen

Got Advanced Data Protection switched on? Then web access is off by default and it'll keep failing no matter what you clear. On your iPhone go to Settings, tap your name, iCloud, and turn on Access iCloud Data on the Web. It may also ask you to approve the sign-in on the phone each time. If you don't have it on, ignore this and go with the cookies route, that's the cause far more often.

that loop is almost always cookies. iCloud's web sign-in needs cookies from both icloud.com and apple.com, and if your browser blocks third-party/cross-site cookies or an extension strips them, it signs you in and then forgets straight away. try a private window first. if it works there, clear cookies for those two sites in your normal window and turn off any privacy or ad-blocking extension for icloud.com. also worth checking the clock on your computer is set automatically, a wrong date breaks the login too.