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...
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.
I gather a lot of schools are adopting ukuleles now. Ukes have a bad name, unfairly I think. (Listen to Eddie Vedder's "Ukulele Songs" album, very underrated). The ukulele is more forgiving than the recorder in my opinion.
It's a fabulous instrument, nuanced and subtle, and you can buy beautiful instruments these days for really not much money at all.
But AFAIK its popularity in school as a teaching instrument has as much to do with Spongebob as it does its capability :-)