I can't answer all of your question, but I can answer some of it.

I highly doubt that slaves were playing classical music - i.e., Western classical music - unless they were taught to play by slaveowners, as slaves from Africa had their own music, the music whose rhythmic language formed the basis for slave songs and later the blues and jazz. By the same token, slaveowners would not have forced slaves or free blacks to play spirituals or the blues - these were musical languages which were developed by slaves and African Americans themselves.

However, and this is most likely the point Miles is trying to make in this passage, African Americans were later restricted to playing "authentic" black forms like the blues or spirituals and barred from playing Western classical music in venues like concert halls. In fact, white composers like George Gershwin and bandleaders like Paul Whiteman (yup lol) were seen as making the blues and jazz "acceptable" for white audiences - "making a lady of jazz," which was seen as a primitive form. Of course, some celebrated jazz for this perceived primitive quality, which obviously is just as simplistic and discriminatory (the "noble savage" concept).

Miles's father was a dentist. It makes sense that, in his ascendancy to the black middle class, he would have been discouraged, or would have discouraged himself, from getting involved in music, because the only available avenue for black musicians was seen as disreputable - vaudeville and Uncle Tomming, or the blues and bars - because the "reputable" path was closed off by racist critics, record companies and audiences.

Artists such as Paul Robeson tried to restore the dignity of black music in America, and of course Miles himself spent his entire career trying to be an emblem of proud black masculinity - wearing tailored, expensive clothes, driving Ferraris, dating/marrying beautiful black women and putting them on the covers of his records, etc. And as a member of the bebop generation who later became (in)famous for turning his back on audiences while he played, he explicitly rejected the vaudeville/Tomming nature of early jazz these later musicians saw in pioneers like Louis Armstrong. And I believe elsewhere in Miles - maybe not, but certainly in interviews he gave - he talks about only listening to classical music at home, not jazz, which is telling.

I kind of rambled here, but in summary: slaves were not coming to America playing Western classical music. Their own musical language evolved into slave songs, spirituals and the blues, which were often used as protest music but ultimately pigeonholed black artists into a narrow, confining musical space defined by a white establishment. Whether it's true or not (and most of Miles is not true, tbh), Miles relating this family myth ultimately serves the purpose of demonstrating how his father had to ignore any musical inclinations if he had any chance of making it to the middle class and achieving any sort of equality with whites, and speaks to the work Miles himself did (following in the footsteps of many others) to rehabilitate the image of the black artist in America.