I agree with others that the public versus private issue varies greatly by location. Your child's personality and level of giftedness also makes a difference. For example, there are several excellent private schools if you live in the Twin Cities metropolitan area, but if you have a kid who is an outlier among gifted, they don't have the best reputation. There are also several different types of public schools that work well with different types of gifted kids: regular gifted magnets, STEM schools, classical academies, arts magnets, and HG schools. Even among HG schools, there are differences. Some are more flexible than others.

For your particular situation, I'd recommend digging deeper than the guidance counselor. Is there a gifted teacher/coordinator for the district? That is where we started in our small city (we are not really in the greater twin cities area!) She helped us tremendously, before DS started kindergarten. We had some private IQ scores in hand, to help convince the school that our child would need something different. When we asked the gifted coordinator for advice, she arranged meetings with the principal and counselors at the school. It helped that our school district did not have an official gifted policy, even though the regular gifted programming didn't start until third grade, because they were receptive to our suggestions and ideas. The principal hand-picked a great kindergarten teacher who was good at differentiation. The teacher did a good job differentiating with reading, and the gifted coordinator ended up teaching higher math to DS. The next year we successfully advocated for a grade skip. Even though we ended up switching DS to an HG school when an opening came up, we were at a point at our local school where we felt that they got our DS and were able to differentiate for him. What was missing was same-age intellectual peers and a faster pace of instruction. That said, if the opportunity did not come up at the HG school, we would have been pretty happy at our local public, because they were willing to be quite flexible.