Extending a narrow high kitchen table with lateral side-wings
I'd build your wings and attach slide-in bars, along the lines of what's shown here. You could use hoop brackets instead of the horizontal bar running between the legs in this example, or go ahead and mount a 1x3 maple or birch board permanently under your table on each side. You could wrap it around three sides so it appears more like a conventional skirt.



image source

This gives you good support, plus you can remove and stow the wings where they won't impeded table use when not deployed. 

Regarding stability, table symmetry is just part of the equation. Table contents and the upper body weight of those resting elbows on the table are probably more significant. However, as long as your table is well attached to the wall at the far end it should remain stable.