Being able to identify, write and play intervals IS music theory. There is no such thing as a "non-theoretical" grasp of intervals. Knowing the sound of and how to construct a pentatonic scale from a major scale IS music theory. Knowing which notes to play in a given key IS music theory. What if someone claimed to be able to do calculations and solve equations with a "non-theoretical" grasp of mathematics? Would that make any sense?
I grew up with a piano in the house, which I refused to learn, but - when I got into jazz 50-odd years ago - I searched out the chords in C for the blues, then songs that I liked. I realised that the chords and intervals in C (and therefore in every key) have an arithmetical relationship to each other, which you can actually see on a keyboard. Later, playing in small "busking" dance bands, where I was expected to play the tune correctly, taught me repertoire. Playing in small busking trad jazz bands on clarinet taught me how to find the right notes not to interfere with the other horn players (harmony). Both of these taught me that most songs written between 1920 and 1950 (the GASBook) often use similar chord sequences (e.g. approx.75% of middle-8s fell into the "I Got Rhythm" or "Honeysuckle Rose" format, and after some time I found I could improvise, generally successfully, on tunes I had never heard before, provided they fell into the broad spectrum of jazz/popular music which existed before, let's say, the mid-50's, to the point that I could anticipate chord changes with a very high degree of accuracy. I do remember having trouble with "Joyspring", but strangely not with "All the things you are", which works wonderfully well mathematically.
I've never studied theory - wouldn't know a pentatonic scale if it hit me in the eye, but perhaps I do instinctively? I do think, for what it's worth, that a thorough knowledge of tunes and an hour or two at a keyboard working them out to find the "shape" of them is time well spent on the road to improvisation. AND, don't take shortcuts like reading off chords - all this produces in most players is a string of ever more tedious arpeggios. Even in good jazz players chord books make them lazy; I played a gig a few years back with a fine young jazz trumpeter who insisted on having his iPhone on a music stand so he could play off the chords, even on "warhorses" that everybody knows. I just kept nudging the stand around with the crook of my tenor in a circle while he was soloing so he couldn't see the iPhone without falling off the stage, so he had to use his ears, provided free by God on either side of his head. He called me a few names afterwards, but he also played beautifully, and with the passion that being out of your comfort zone engenders.