My workaround was to go to the actual OneDrive folders on the PC, right click and then select 'always keep on this device' in the menu.
The status icon should then change from a cloud to a white tick in a green circle - once it's downloaded.
Granted it's not an elegant solution, but it worked for me.
Confirmed, OneDrive in "Personal mode", currently at version 20.084.0426.0007 (installed on Windows 10 Pro, version 2004, build 19041.329). I tried reset of OneDrive, reinstall of OneDrive, registry keys mentioned in this thread. Nothing helped. Checkbox will never accept "unchecked" state.
It looks like someone in Microsoft decided to force this as the only option, but forgot to remove this useless checkbox from UI.
The only thing that seems helpful is this workaround with "always keep on this device" which needs to be set explicitly per folder. Thank you, @billallen_UK