Yes, it does. This is the second 4001 I have seen with one. The first was the first 4001 (a '72 or early '73 FG) I ever got to play that was owned by an old friend in high school. The MG 4001 is from March 1973 and is currently with a friend of mine. When I found the bass I thought that it might have been Gary's that I played in HS and refinished, but it turned out not be that one (the MG finish is the factory original), so, at least, two early '70s 4001s have them.
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For me, it’s not really a desirable feature unless you’re a collector. I knew people that would unscrew the snaps. I didn’t like the pads on Gretsch instruments. Being vinyl, they tended to crack and tear.