Best/cheapest way to get 14-bit adc readings with Teensy (14-bit MIDI)

I'm aiming to build a high resolution (14-bit) MIDI-over-USB controller using a dj crossfader, and am wondering what the best way to get the actual hi-res adc into an Arduino?

I was going to use a Teensy as I've used them plenty before for MIDI-over-USB purposes, but the LC lists a "usable range" of only 12bits, and the regular Teensy lists a usable range of 13bits:
https://www.pjrc.com/teensy/teensyLC.html

Since I'm going to be controlling gain stages in software, I'd want as much resolution as possible, and the difference between 13bits and 14bits is pretty big (8k vs 16k steps).

There are also dedicated 16-bit adcs:

But then that adds another layer of complication/cost/size.

Are there any "native" "true" 14-bit adc Arduino boards? Or can anyone think of, and/or suggest a different approach here?

That $15, 16-bit, 4-channel, I2C ADC breakout board looks like a good match for your needs. Is there some reason NOT to use it?

Not especially, I just didn't know if there was a 'native' 16-bit solution.

Space is a little tight, as I want to fit everything into a little abs enclosure, but this would also let me try to do 16-bit over three MIDI CCs for even more resolution (7 most significant bits, then 7 least significant bits, then 2 even less significant bits). I've never done that, but I wouldn't imagine it's more difficult than just bit shifting the appropriate bits over on the other side (Max/MSP).

I'm aiming to build a high resolution (14-bit) MIDI-over-USB controller using a dj crossfader,

You might want to do some experiments (with software) before you build a 14 or 16-bit hardware.... I mean some listening tests to determine how many "steps" you need.

Although you want 16-bit (or 24-bit) audio resolution you don't need 16-bit *volume-*resolution. For example, most digital pots are 8-bit and that's better than you can do with a manual-analog pot.

And, if you feel it's necessary you can add some digital smoothing/filtering between steps.

P.S.
Since you are the programmer you also have the option of making constant-dB steps. For example, with 8-bits and 1/2dB steps you'd have a range of over 100dB.

Normal 16-bit PCM audio is linear, so "one step" at the low-quiet end from 1 to 2 is +6dB. But at the high-loud end, one step from 32,766 to 32,767 is +0.000265dB.

For general purpose stuff, I agree with you, in that the resolution I'm talking about is actually imperceivable. (12 vs 13bit)

But I'm working with audio feedback, and imperceivable changes in volume can have a big impact in a wildly dynamic (feedback) system.

This is the kind of stuff I'm talking about:

Although you can't see it in the video, I'm using an analog volume pedal in the signal path, as that gives me infinite granularity in terms of the gain stages (not that I need infinite steps), whereas MIDI steps would be useless in that regard.

Once I get the numbers in the system, I'd apply some scaling to something more usable (obviously some kind of audio tapering), especially since dj crossfaders are super nonlinear. There's generally a really sharp cutoff at the end, and then it's more or less 'full volume' after that. Since I'd be working in software I can be more nuanced with that non-linearity and have a steep cutoff (with a specific curve applied), then a slower increase over the rest of the range (at another curve), etc...