I think showing the menu automatically is great, just that when I go to re-open it myself, the button at the top right does not feel natural. I know there was a panel there, so my natural tendency is to scroll or move the mouse cursor in search of it.
You've only got a few core features, so you've got more creativity available in how you present them: On desktop you could reveal the nav when the mouse cursor is in the top 100px, and on mobile show it when the user scrolls to the top. No clicks or taps necessary, because there's nothing that would conflict with that behavior. Consider something similar for the player/info panel at the bottom. When the user stops interacting for a second or two, slide the panels away and show only the visualizer.
Curiously, I don't normally click on the hamburger icon because hitting the ESC key also toggles the top menu, so I completely agree with you that it doesn't feel natural.
I've been tempted to put more information on the player panel but have forced myself not to do so to keep the UI as clean as possible (mainly thinking in mobile browsers). I guess I could put in some icons, though, that would give hints to user as to what's actually going on behind the scenes.