Check the road force/uniformity of each mounted tire and wheel assembly. You can balance an out-of-round assembly but it'll still vibrate.
If the vibration comes and goes with varying speed (harmonic-related), it is almost certainly related to uniformity/out-of-round problems. Any decent tire shop
should have the right equipment to do a road force check although it is probably related to flat spotting, as you mentioned, if you had no
problems before it sat.