I've not completely absorbed the delineation between a window manager and a desktop manager, but I believe that LXDE is the culprit, either directly or indirectly, in conjunction with pcmanfm.
I've read that installing pcmanfm-mod might solve my problem, and I've also read that pcmanfm is no longer being developed, and is replaced by spacefm, which is out of my scope to install right now, as it is not available in my distribution.
To bottom-line it, XFCE with Thunar has solved the problem. I added a "Launcher" to my newly-installed XFCE-Panel, and its EXEC is identical to the one in the original post, and the %u is now successfully passed to my script when I drag a link from Firefox. I have not tried to drag a link from Chrome, but I feel it would work as well.