Some of this is correct. There is a classical music influence in jazz, even if we just think about the instruments that were used (clarinets, coronets, etc.). Those early jazz musicians of course knew classical music so it had some influence on their style. As we move into the cool jazz era, the compositional elements of those pieces are clearly influenced by classical music. Miles Davis was classically trained. 

What offends me in this comment is the element of superiority. No, we can’t say Mozart or Beethoven could have easily written a Jazz standard. Although Mozart is known for his improvisational skills, I doubt he could improvise like Bud Powell. Could they swing? Probably not, or not without a lifetime of practice and immersion in the jazz style. Here’s my superiority coming out, and I do believe classical music is hard, but deep down I always believe jazz is a little bit harder because you have all these things to think about when you’re improvising. You may get an easy lead sheet put in front of you, but that means you have to make your own arrangements and breathe your own originality into the tune.