I agree, it totally depends on your child and family. Some kids do just fine staying home with Mom while others need the extra social interaction. My oldest loves being home with me but also needs his time away to be his own person. I have to admit that while I am a stay-at-home mom so I raise my children and be with them as much as possible, I enjoy the break, too!
When my oldest was 3 we did a mom-run preschool where a few other moms and myself took turns teaching. My son loved it! He was SO sad when I decided not to do it the next year (I was pregnant, sick, and had a broken ankle) and begged me to send him to preschool. If we could have afforded it, I probably would have done one that was just a couple hours 2 days a week.
We put him in a preschool this year since he just barely missed the cut off for Kindergarten and he is in HEAVEN!
Academics- with the exception of numbers, the ABCs, and writing his name- really should not be an issue at 3-years-old. We are doing it for the other reasons you mentioned: learning to take direction from someone else, learning independence, and practicing social skills like sharing, taking turns, etc. Since my boys are more than 4 years apart, it's good for him to be around lots of other kids his age.
I don't know where you live, but if you are seriously considering preschool for your 3-year-old I can recommend a fantastic one in New Hope.