My kids are ds15 1/2, ds10, ds7, dd 4 1/2 and I work pt most of the time. I think I understand where you are coming from.
You can see that things at home are not the way you would like them to be, you feel you are not meeting your dc needs.
IME I take the most obivious thing to be rid of when I feel overwhelmed and want to delegate some of my life out to others. But for us I know not homeschooling would actually make life more chaotic
But sending dc to school might help with the social thing, it might not and might make things worse for her. It won't make your life easier because PS comes with its own schedule and foibles to make your life more interesting,
it won't give you more freedom if you are still at home in a house kinda out of control. Maybe not look at it as having to 'do it all',
but break down what doing it all means to you and figure out what is going to make you feel more in control of your family life, so you know which thing to conquer.
I don't see this as about homeschooling, more about being a mom with young children and finding the groove or path for your day to day life that results in all of you in your home happy and healthy to enjoy the days.
It sounds like it is time to regroup and build anew the next year.
I would start by establishing a routine for the day and setting simple goals for myself and I would stop to examine what I really want from my family life.
Look at yourself thinking, do you need to go get a haircut? taking care of your self personally, are you eating right and excersing? got any fun things you like to do hobbies, a positive mentor mom df you talk to frequently?
Look at home with an honest eye, how much stuff can you get rid of? If your dd hates clutter, start in her room. Can you clear out 1/4 the stuff in your home to donate to charity & use the receipt for tax write off next year? Can you find the money to get a cleaning lady to come in and scrub the heck out of your bathroom and kitchen? Do you have your freezer stocked with simple things to pull out to put together soup and sandwich quickly for no stress dinners?
Do you have a central calendar for kids activities that is up to date? Do you belong to an active hs support group and are meeting with other moms at least once a month and you dc gets to play with other hsers at least 3-4 times a month? As far as meeting social needs, decide how requently your dd needs contact with others outside the home and then you can find creative ways to do that. invite others over if you can't get out. Ten minutes here and ten minutes there is normal direct instruction time for families with young children and that should be sufficent to meet your childs academic needs.
It is okay for your ds to nap in a stroller at the playground or in the car taking lil two naps while you drive to the next town for dd to be in 4-h or scouts. And don't beat yourself up about your dd, she will be fine. You have the whole summer to reinvent, I would wait on thinking about school till closer to fall and right now focus on yourself and your needs.
I am in so cal too. ((((HUG))))