This is a confusing and difficult situation. It seems to involve at least two issues: you do not feel competent to teach particular high school subjects to the extent that your daughter will be well-prepared for college, and your daughter wants more social interaction in her life.
Teaching high school is not as easy as teaching elementary and middle school, and the pressure is on to prepare for college as you are already aware. There are online and video courses to help if you do not feel comfortable teaching certain subjects. Please keep in mind that some certified high school teachers don't always feel confident of their ability to teach either, so you are not alone. I myself have spent years writing and teaching writing, and sometimes I get stumped on how to teach a problem area.
Regarding your daughter's desire for more social interaction, some children and adults seems to need more opportunities to socialize than others do. I would be a perfectly happy hermit, but my son would not thrive in such an isolated situation. In addition, your daughter may simply want a taste of what high school is like. She probably does not have anything to compare with her present experience in that regard. You have a wider view of the situation, but she is not going to be able to understand the current situation using your wider viewpoint. She can only judge and hope based upon her present feelings and desires.
You may decide to let her try a year of high school and see what that is like. Although you do not know what the high school is like, you could probably visit the school or make contact with someone who does. The private school I teach at often has visiting families and visiting students who are trying out the school for a day or two to become acquainted with the atmosphere. That is a possibility, and once she's had that taste, your daughter may decide that she doesn't want to be regulated day after day by bells and clocks. The crowds and constant rush may be more than she expected or imagined. On the other hand, she may enjoy it because of the variety of social experiences and the opportunity to be part of a group.
Regardless of whether your daughter goes to a public school, you can still home school her. She will continue to need your love and support and guidance in different areas. Perhaps this summer, you can talk with your daughter and look at some options. The more she feels involved in the situation, the easier the outcome may be for the whole family.
God be with you as you work through this situation.