I figured out what the problem is: always-on-top. If you set the topmost option, then the buttons do not appear on the taskbar player. This is true whether the setting is saved or not (that is, even if the player starts without the on-top setting and the buttons are visible, simply clicking Video->Always on Top turns it off and keeps it off until you restart VLC; even turning it off again won't bring the buttons back).
That's unfortunate because the topmost option is necessary and it seems odd to have to choose (in fact, it seems odd that there would be any practical reason for them to have anything to do with each other, so I suspect it's just an interaction bug).
Old thread, but still mine, nonetheless.
Dead-on. Always-on-top setting is always on for me. I turned it off and the buttons immediately presented themselves when the was program reopened. I'm up to 2.0.7 x64 at this point.
Turned always-on-top back on and the buttons were gone again after restarting the program.
For the most part, leaving always-on-top enabled for the past three years, the buttons populate about 75% of the time. They seem to disappear when the program is accessed after a fresh update installation and then they're randomly missing every so often during regular use.