Ideological dissonance, and a case for electronic 'ocarinas'
The ocarina is often marketed as 'easy to play', and in some regards it is. They have an approachable fingering system and it is easy to get a sound out of an ocarina. But in other regards playing this instrument is technically challenging. There are many hidden details, and intuition often leads to poor solutions. There is an ideological dissonance between how the instrument is presented, to some extent how it presents itself, and what it actually is.
Now, this marketing approach works as there is an element of truth to it, ocarinas look simple, and a beginner can quickly play something that they recognise. But long term this approach tends to throw a road block in the learners path. To some extent this arises due to the open-ended nature of instruments, but I do think it can be reduced. An electronic 'ocarina' could take the elements which are intuitive, like the fingering system, and automate the ones that tend to cause problems, reducing the traps for a learner.
For instance, an electronic 'ocarina' could take the 4 hole system, which is often intuitive for children, and eliminate the tuning errors which are inherent in this design. If blowing pressure where mapped to volume instead of pitch, you'd have an instrument that's much easier to play in a group with little experience, while still achieving a good sound. Such an instrument can also be played on headphones, and thus a learner never has to worry about bothering others. By comparison, acoustic ocarinas are loud, and attempting to mute one mostly makes it sound bad.
I don't think an electronic ocarina would have to be limited to being a teaching tool, though. From the point of view of a serious instrument there are many aspects of the ocarina that are challenging to deal with. For example, an ocarina's sensitivity to pressure varies a lot over its range. Due to this, creating a consistent vibrato requires a much larger pressure change on the high notes than the low notes. Having two thumb holes is also an ergonomic challenge, and the limited range constrains what music can be performed.
If the pressure non-linearity was reduced or eliminated, the range extended—perhaps by emulating overblowing found on other instruments—and the right thumb used only for support, the task of playing would be much easier. While you may think this 'dumbs down' the instrument, it does not. People only have a limited number of 'cleverness beans', so if the fundamental task of playing is easier, the skill ceiling is raised.
It is also worth considering the flexibility allowed by electronic instruments: pressure could control pitch, volume, or an arbitrary mixture of the two. Pressure sensitivity could be raised or lowered to account for preference, or to accommodate new players. The design flexibility could also eliminate the common ergonomic issues experienced by children playing an instrument designed for adults.
This is not a new idea, and there are multiple examples of electronic 'ocarinas' made as hobby projects. They utilise capacitive touch pads and allow things like pitch slides to be performed, depending how much of the pad is covered. I do think this can be developed much further than it has been. If enough effort were put into this, I can see it becoming a very able instrument.
The ocarina's tone is close to a pure sine wave with few overtones and a bit of wind noise. I don't think it would be that difficult to synthesise realistically. I suspect the main challenge would be modelling of details, like the slight changes in sound caused by how a note is tongued and the pressure at the instant the note starts. Electronic instruments in general tend to handle such details poorly.
I suspect that poor handling of these things is at least partly due to MIDI, as it is based on discrete notes. The ocarina is essentially a continuous pitch instrument and I think that it would be better modelled as such. Another factor is that less information is available. Acoustic wind instruments are sensitive to hole shading for example, whereas capacitive touch only detects contact as far as I'm aware.
Details of that kind can be simulated, although I don't know if there is a commoditised(cheap) way of doing so. An experimental project by Google called 'Soli' detects hand gestures in free space using miniaturised radar, and something of this nature would be able to detect the distance between a finger and a simulated 'hole'.
As I already have a lot of time invested in developing acoustic ocarinas I have no intention of developing this idea, thus it is left as an opportunity for the reader. I suspect that even a basic electronic 'ocarina' would be easier to market along the 'easy to play' idea, as it would be closer to the truth. It is already known that this idea works in principle, as Zelda: Ocarina of Time does essentially the same thing with its virtual instrument.