I understand the intellectual satisfaction you're talking about. Now I'm wondering if we can push the minimalism to using these four holes with only two fingers. It seems doable from the diagram you link, using link fingers, and assuming half holes can be covered from the left side, and two holes can be covered with one finger at the same time. Would that work, or does the physicality of the instrument prevent that ?
Hi, Ocarina maker here! This is a genuinely interesting idea, and some Ocarina like the Goslings do have a double keying with a single finger.
Sadly with English Vessel Flutes (4/6/8 holes) the holes need to all be roughly equidistant from the center of the chamber, which makes a two finger setup ergonomically complicated while keeping a tangential fipple. It is also very desirable to use the pads of the fingers to cover the holes, making a good air seal is tricky with the middle of the finger. On ergonomics, it is also worth mentioning some Ocs get quite big as well! My largest 6 hole is almost 300mm wide, my fingers are not quite long enough to span this...
From memory — can't find the thing right now! — this would be a little difficult on the one I had because of its shape.
Also because — not sure how to explain it — but in my experience the holes on a four hole ocarina are also sort of finger grips, in a way. That is to say, you support the ocarina against your lips with at least one finger below it and two thumbs behind it, but part of your grip on the instrument is gently transferring from fingertip to fingertip as you play, a bit like a recorder or the pipe of bagpipes. The "open" low note has the loosest grip, and you might even subconsciously tilt your head back slightly to allow the weight of the ocarina to shift more to your thumbs.
It might be possible to design, effectively, a one-handed playable instrument that works the way you are talking about, but I think it would be quite uncomfortable.