Why? I was part of an excellent gifted program in a public school so I know it can work. I'd prefer to offer enrichment at home in LO's primary area(s) of giftedness.
And I have a lot of negative experience with homeschooling (15+ cousins and family members), some of whom where gifted in various ways, that resulted in very limited educations and poor social skills.
I've seen homeschooling well done and poorly done and I have seen some instances where it results in limited social skills so I do understand that end but it doesn't have to be that way.
I am assuming that your "why" is addressed as why I needed to make significant accommodations and why I feel that the GT programs are a better fit for bright high achievers than gifted kids. I'm running so I'll be brief, but basically we've seen a dumbing down of general education to the point that many families want more for their children and many children who are not technically gifted can do more.
Our local school districts wind up placing about 15-20% of the kids in GT or accelerated classes. That leaves kids in the top 1-2% standing out like a sore thumb in those placements and not having the classes geared toward their needs b/c they are still outliers in that setting. My kids are also HG and one is 2e. Both of them started early and one has skipped a grade and has many areas where the work has still been more about large quantity than appropriate level.