There are appropriate teachers for even the most talented child prodigy in the greater Boston area, because there are people who play in the Boston Philharmonic and the Boston Symphony Orchestra, and people who teach at NEC, BU, Boston Conservatory, and NEC. Through this wealth of musicians, it definitely should be possible to find the right teacher, if your child is outgrowing their current teacher. In addition, the program at NEC or at BYSO would provide the right small and large ensembles. Another thing to consider is music camp - the right music camp can provide an excellent peer group setting for a young very advanced music student; for my kid, it was really the only place where they felt at that age that they could be completely themselves in a peer group. We didn’t find music camp until the kid was 11 or so I think, and it made a world of difference for them, being in a setting where all the other kids “got it”, how important and fun music was.
The reality is that no matter how good a musician the child is, the most important thing is to balance normal social development with the many hours of practice required to make a professional classical musician. Your child needs normal school, normal friendships, normal outdoor and indoor playtime with peers, unsupervised and unorchestrated by adults. This is tough to achieve with even the only 1-2 hours of practice a day required of, say, a brass player at that age; it is extremely difficult to achieve with the more like 2-4 hours a day of practice required of a strings player at that age. Add in having to travel 8-10 hours roundtrip for an all day Saturday program in NYC, and I’d say that that child would virtually never be able to play with friends or go to children’s birthday parties.
Bear in mind that someone who teaches at the college level may not be the right match for a young child; in fact, most who teach at the college level refuse to teach children, unless the child is an exceptionally mature, exceptionally motivated, obviously prodigiously talented musician.
I do not think that you need to move to NYC now for Juilliard pre unless you seriously have a child who is already a recognized prodigy, and even then, the appropriate teacher should still be available in the Greater Boston area. It is extremely unusual for any child to be having three lessons a week - to me, this is a red flag, unless the parent is willing and able to pay for what is essentially serving as accompanied practicing sessions (which I did myself for all my kids, kept them company while they practiced, less and less as they got older, completely stopped by about age 11, except for an occasional suggestion that “Now would be a great time to practice”.
Sure, if you feel that it is appropriate for your child, go ahead and audition for Juilliard pre this coming spring. They will tell you if it is the right setting for your child. But I wouldn’t commute to it from Boston, nor would I move there unless it is the right thing for your entire family to move to the Greater NYC area. If it is not, I’d focus now on finding the right setting in Boston for your musician.