Hello,
You are right. On office.com there is no single Clear all for Quick access or Recent today. The supported way is to remove items from your view.
Sign in at office.com, go to Home or My content, hover the file card you want to hide, open More options... and choose Remove from list. You can select multiple items with their checkboxes and use Remove from list to do this in a batch. This only clears your view, It does not delete the files.
If you actually need the files gone, open the item's location in OneDrive or SharePoint from the same menu and delete it there. That removes the file and it will disappear from Quick access and Recent after the page refreshes.
If anything in your tenant looks different, tell me which section you see first on sign in, Home or My content, and I will adapt the steps.