OS X Mavericks :: None Of USB Flash Drives Can Be Accessed In Finder

First the specs from "About this Mac":

MacBook Pro "13-inch, Late 2011", 2.8GHz Intel Core i7 // RAM= 8 GB 1333 MHz DDR3

Mac OS X 10.9.4 "Mavericks" (13E28) 

The problem:

None of my USB Flash drives are accessible in Finder. 

The drives do technically "Mount" properly - they show up in Disk Utility, with the "Unmount" & "Eject" buttons available (indicating it's mounted), and they're mounted to the path /Volumes/<Disk Name>.  Also, the disks show up in the Finder Sidebar with the correct name etc.

However, If I click on the Finder icon for the drive, the window flashes white for a millisecond and then reverts back to whatever the window showed before (eg. my home folder's contents).  Finder will not access them.  This has occurred on 3 different USB Flash drives (all of the ones I own!), which are formatted for ye olde windoze, FAT16 or FAT32 (aka. MS-DOS).

The drives operate perfectly normally on an iMac running OS X 10.8.5 (accessible in Finder). I was able to copy files to/from & list the contents of the Flash drives normally. 

In Disk Util. I can Unmount & re-mount without error.  If I unplug the disk I get the expected "This disk was not unmounted properly" banner.  Eject via DiskUtil appears to work properly without error, causing the disk to vanish from Disk Utility (as expected).

The directory /Volumes does not show the drive's directory (just "Macintosh HD") - in fact the first time I list /Volumes/ after plugging in a drive, it appears to show some folder in /Volumes that is quickly removed (kind of scrolls-up into itself) as soon as the /Volumes directory is listed - was it my drive playing hide-and-seek??

The Eject button / option (in right-click menu) is not available in the Finder.

If I use Finder > Go To Folder… and type in the "Mount Point" from Disk Utility, I get a "folder not found" error. 

In DiskUtil, if I right-click the drive name and select Reveal In Finder…, strangely the folder will actually show up in /Volumes, and I can open the folder and view the drive contents (sometimes I have to Reveal it twice for it to work)!  I was also able to create a new folder after this (which did show up on the iMac as well).  However, if I tried to copy and files to the Flash drive, I get either

The operation can’t be completed because one or more required items can’t be found. (Error code -43)

(when I try to drag a file into the newly created folder on the Flash drive), or

The operation can’t be completed because an unexpected error occurred (error code -8058).

(if I drop a file on the Flash drive's root directory, or try to copy a file form the Flash drive to my computer). 

I have tried DiskUtil/"Repair Disk" on the flash drives, which doesn't show any errors or warnings (they appear to be ok - again they work fine on another Mac).

I also did a "Repair Permissions" on my MacBook, to no avail. 

I seem to remember having this same problem on OS X 10.8 at some point last year, and think I solved it via an online hint, but I have yet to re-find that solution (I'm on OS 10.9 now). 

Info:
MacBook Pro, OS X Mavericks (10.9.4), USB Flash Drives formatted as FAT16