The way I'm reading this, you have three options, and two of them are pretty good ones. That's a nice place to be.
If the public school won't properly support your DS until 1st grade, then it makes sense to eliminate them from your consideration until at least then.
My DD's profile was very similar to what you're reporting here at this age... she was encouraged to show her talents in preK by a very understanding teacher, was highly-respected by her peers as a result, and had a wonderful time with the social aspects of that, because she's a highly-social child.
Then came K, in which she had no support whatsoever for her talents in public school. It was a disaster. We had no private school option that would properly support her, so we homeschooled.
My DW had the same reservations you do about her own ability to teach. We also had the same concern about DD's social-seeking needs. And I can tell you that in our case, homeschooling worked out beautifully. In fact, fast forward a few years, where DD has finally gotten the grade skip she needed, and access to a G/T program for half her school day (which she adores), she's fitting in well with peers, and yet she STILL says she'd prefer to be homeschooled. DW's fears were unfounded... DD thrived in homeschool.
The approach that we took with the school over the years was to let DD try it whenever there were some new accommodations that might make it work out, keeping the homeschool option as a fallback for the many times when it didn't. That might be a good strategy for you to use with your DS' private school... let them try their interventions for your DS, and see what happens.
