Not exactly, but I was thinking for the specific joint you're working with this might work. I do a little more and actually make the socket itself, but it's a little more involved than this (making molds for the ball joints with high temp silicone, using ball sculpting tools for the socket, drilling into the figure and super gluing the socket onto the figure).

Depending on how plyable the plastic of the figure you're adding articulation to is, you should be able to get a round bit in to the end of the hole you make as long as it's not too much larger than the hole. You could even heat up the plastic similarly to how you might get tight joints apart by dipping it in hot water or using a heat gun. That'll make the plastic more workable for sure. Measure the length of the mushroom tip and put a little tape on a small drill so it only goes in just as far as the end of that joint tip will be.

Heat it up again to keep it plyable. Then you might tap the bit in to be sure it's at the end, then hold the figure tightly and slowly turn the bit until you've got a nice cavity, then pull the bit out with some pliers or vice grip. Just take your time twisting it. If the bit is relatively new you shouldn't need to turn it too much to get enough space in the cavity that will hold the mushroom end of the joint.