Hey there VC-J,
Thanks for reaching out to Apple Support Communities. We understand you’re having to repeatedly trust your computer to access iCloud.com. We’ll be happy to help.
Do you have any security software running or firewalls on your PC? Are you logging in through a VPN or captive network? If so, this could be one reason this is happening. Make sure your computer is up to date and there is no security software or firewalls running and see if this has any impact on the issue.
You may also want to consider using iCloud for Windows to access your data as well. You can learn more about this here:
Set up iCloud on your Windows PC
We hope this helps.
Take care.