Yes there are plenty of examples on youtube of people playing all kinds of classical music on the Hammond Organ ,I play some classical music on my Hammond Organ,the idea that the Hammond Organ cant be used for classical music is absurd.

A year after I graduated from high school, I was invited to play for commencement. It was held in the civic center that had a Hammond B3. If literature is carefully chosen, it can work quite well. Obviously with just a 16' and 8' draw bar in the pedal, you cannot play a solo melody in the pedal. And while you can make the pedal loud, it does not do so well with a fugue subject. However, there are opportunities to solo out a line. Drawbars can synthesize a clarinet, oboe, trumpet or cornet quite nicely. I remember playing C. S. Lang's Tuba Tune in D Major and it worked quite well.

Back in the eighties classical was even played on Yamaha electones, I'm glad to be past that though.

Some years back I bought the C3 which belonged to the church I went to as a child... it had SO MUCH classical played on it in the 60's and 70's. Very nicely. Beautiful tone.
... it is now a very tired, old and abused C3. The abuse, tiredness and aging came AFTER its heyday though, when it became a potplant stand at the back of the church and had people poking the valves/tubes into all the wrong sockets- causing a power supply meltdown in the PR40 amp. I add that it never had the experience of a leslie, it was always coupled to its very staid but richly toneful Hammond amp.
Organ lessons for me were all classical because that was the culture I grew up in... and it taught me a lot which I took into a contemporary setting. (In other ways the set piece learning style retarded my musical development. There are teen keybd players at church who understand music in ways I didn't uncover unassisted for decades!)
I admit I haven't touched a piece of sheet music for thirty years now, let alone played set classical pieces.
...Aside from a bit of baroque of course, cos' Bach crosses ALL musical styles!
The motifs from the great composers show up in all my playing on all my instruments (mainly electric guitar these days), but not often in full.
The Hammond is still mighty fine for classical players, it WAS designed for traditional churches first and foremost.

I don't know that I've ever played anything but "classical" music on my Hammond.

I only play music I have written myself on my Hammond, which is a jazzy rock kind of thing but if it helps I play it very seriously.

An elderly gentleman here who passed away about 10 years ago was a widely-known classical organist and church musician in his younger years. His name was Edward Linzel, and he was actually a much-less-famous compatriot of Virgil Fox and E. Power Biggs. He used to tell me little stories about a variety of escapades he had with those two when he lived in NYC 60 or 70 years ago. Anyway, he was quite a fan of the Hammond organ, at least before digital organs became really good at simulating pipes. While most of his career had been spent playing great pipe organs, he had always preferred to practice at home on a Hammond because he felt he could get more of the nuances of organ tone with the drawbars than with the somewhat poor imitations of pipe stops we got from electronic organs prior to the maturing of the digital organ.

I also remember that in my earliest days in church music, back in the late 60's and early 70's, there were "experts" who led the occasional music conference I'd attend who adamantly proclaimed that the Hammond was an "authentic" instrument in its own right; if not really great at simulating pipes, it was a truly unique music maker that could be artistically registered and played. These same "experts" were often quite dismissive of the run of the mill oscillator organs, considering them tacky and inadequate impostors that pretended to be suitable substitutes for real pipe organs. I thought this attitude was a little odd even back then, as my ears told me that the really good oscillator organs were pretty darn pleasant to play in the absence of real pipes, and I was less than sold on the Hammond because I loved the crisp articulation and "chiff" and such that electronic builders were beginning to put on their organs at the time in response to the "Baroque revival" trend in organ sound.

Thus I never became that much of a Hammond fan myself, and even today I am a little annoyed when I have to play one that is noisy and hummy, though I have learned to better appreciate the interesting sounds one can get from the drawbars.

Hammonds are quite versatile and certainly can be used for the classical repertoire. This is especially true of the models with an AGO pedalboard and the pedal solo unit, such as the RT-3. When I was learning to play I did all of my practicing, including classical repertoire on the Hammond. Where Hammonds are lacking is in the lack of credible ensemble due to the method of synthesis and single bank of tone generators. The pedal solo unit, being a separate rank of oscillators, helps a bit with this as well as overcoming the tubby pedal tones from the drawbars.

Here's an RT-3 with a Leslie cabinets in an ideal acoustic setting playing Franck.

Isn't it on IMSLP?
And yes, either an ideal acoustic space, or a good reverb is a must IMO to play the classical repertoire on a Hammond.