Indeed, thanks for addressing the original question! The following is just observations for my system -- no real insights or fixes. And I'm not asking for any help, so this is easily ignorable if you're not into endless minutiae.
I "use" OneDrive all the time to keep my computer backed up, but only in a passive manner. That is, I never (overtly at least) do any of the OneDrive things that people have talked about here -- like "connect", "sign in", "log in", etc. As far as I can tell, whatever is needed happens automatically, and everything just keeps chugging along in the background, without my involvement. And I've never had any problems, like being locked out. However, the brown boxes are only one of the adornments I have seen. Mine cycle thru at least the following icons (on their own, so far as I know): green circles with white checkmarks in the middle (I assume that's the best), brown boxes, 2 blue arrows in a circle chasing each other, and no icon at all. (A cloud icon was mentioned above -- if that ever applied to me, it was too long ago to remember.) And it appears (without exhaustive searching) that all my folders, subfolders, and files have the same icon at any point in time. If there is a particular order to the cycle, or a particular periodicity, I've never kept close enough notes to notice. But thanks to Patrick Powers above (3 Mar 2023), I have found that right-clicking the desktop and then clicking Refresh converts the brown boxes and the blue arrows into green circles. (Don't know about the no-icon case yet.) And the conversion appears instantaneous from a human's standpoint, which suggests the software is not going to all the thousands of my files in my OneDrive cloud space. But I'm just seeing the top level, so perhaps non-trivial time is spent going to all the subfolder depths and files therein, which I don't see.