This table was incredibly shaky when it came through my door. This is a common problem with older pieces. The glue in the joints had broken down and was turning to dust. There is really only one good solution to that problem. No, I didn't use screws to reinforce the joints.
I needed to completely disassemble the table -- carefully breaking all the glue joints. I cleaned them out, removing all the remaining glue, and then re-glued it.
Pieces of veneer had been missing from the surface as well as the inlayed flowers. Lastly, one piece of the apron was missing, so I had to carve a new piece to replace it.