I want to get the chewed-up insert out of the socket, but there's next to nothing to grab hold of, and what's there breaks off too easily. I'm pondering taking a Dremel to it to cut some grooves for a slotted screw driver to grab (what's left isn't deep enough), but I'm wondering if there's a better (read: Less likely to mar the base) way.
Nothing to grab? These reducers have slits to put the blade of a screwdriver in.
And at yours I see those slits too.
I these are worn out, cut at different place new ones. With a hacksaw, but cautiously.
Alternatively you can cement an apt screw into the reducer and by that twist it out.
There is a device called an Easy-Out (or generically, a screw extractor) that is a shallow left hand thread on a taper. Such a beast of suitable size is placed into a hole in the piece, usually after drilling the center of a broken screw or bolt in the normal situation, in this case you already have a hole. The Easy-Out is then turned counter-clockwise and the more one torques, the tighter the left hand thread grips the inner surface. They have a short square section on top that can be gripped with a tap wrench of suitable size. The threads of those adapter bushings are so near each other in diameter there is almost no metal in them, so preventing damage to the female camera base thread might be tricky. Interesting problem! (But sounds as though you've solved it).
Standard camera sockets are 3/8-18 and 1/4-20. The difference between the two is 1/8 inch or 1/16 per side. A screw extractor/easy out whose tip will fit snugly into the 1/4-20 insert without bottoming in the socket will be quite large and expensive.
One of these will work also https://www.amazon.com/Extractor-Ai...5&sr=8-1-spons&keywords=screw+extractor&psc=1
A wide blade slot screwdriver will work and you may have one on hand. I have had success with one. The key is to keep the pressure even on both sides and not to drive the extractor in too deep as that will wedge the insert in tighter.
Screw extractors are designed to be used in holes drilled by the repair person. Thus he can adjust the lenght and the width of that hole corresponding to the shape of the extractor.
With a given hole width and depth typically a screw extractor will not fit inless shortened.
I'm not familar with imperial sizes, so it may be that by accident the tip of one extractor just fits that given hole though.
The advice I give here is based on a non-mechanic without special tools.
Use a 1/4-20 bolt with a nut threaded on it. Run the bolt into the socket until it bottoms, then back out 1/4 turn. Then run the nut up the bolt until it tightens against the insert , then back the bolt out. It should bring the insert out with it.
Use a 1/4-20 bolt with a nut threaded on it. Run the bolt into the socket until it bottoms, then back out 1/4 turn. Then run the nut up the bolt until it tightens against the insert , then back the bolt out. It should bring the insert out with it.
That's likely the best / simplest / easiest / cheapest way to do it.
It might benefit from putting an internal-star lock washer between the nut and the insert.
That will increase the torque on the insert when trying to remove it.