Naturally JI instruments include any instrument which has a continuous frequency domain, like trombone, fretless stringed instruments, and human voice.
Instruments with keys or frets may be "bendable". However, the direction and quantity of bend varies with with each tone it is paired with, making this incredibly impractical.
The "beating" that occurs in 12TET is inaudible to most western listeners because we've become so acculturated to it. You have to get some exposure to music where it doesn't happen before you even notice it (and thus seek out more consonant harmony).