This already happened once and I removed the panels the legs are attached to (from the table), and found that the legs have basically a single large nail holding it to the panel (looks like a railroad spike or something). I hammered it down, but the legs weren't completely flush with the panel and still had a bit of a wobble. Then all I did was roll it across my tile and the grout dip was enough to pry it back out (the picture shows after my wife also spun the table, and the leg pry'ed more tipping the table over).

Any idea what the "correct way" is to fix this, and get the legs flush against the panel, and how to prevent the nails from prying out in the future?

Tempted to just add some wood screws and wood glue between the panel and the legs, but I don't want to damage a family heirloom.

Your problem is physics: Those legs are just a long lever, and there is nothing to prevent sideways pressure on the bottom of the leg from bending that nail.

Most tables have sideways bolts attaching the legs to an apron to keep this from happening, or a trestle towards the bottom to link the legs together.

This is going to take some major surgery to add better fasteners to the tops (a threaded insert with a bolt that goes through those boards), and adding an apron around the legs to give them sideways support.

If all you do is add the threaded inserts, the next time this happens it will splinter the leg because the bolt will be stronger than the wood.

Anyway, sounds like more work then I'm willing to put into this. Thanks, but I'm just going to replace it.

Thought it was odd that this design was supposed to work.