Ha! Actually my kids had even more socialization. We did the public school, private Christian school and then (upon their request) Homeschool.
IMMEDIATELY upon starting Home School my kids self esteem shot skyward and they met a large group of kids (some Christians, some not) that they became leaders of. In Al, we are satellite teachers of a private church school umbrella. We met every Friday and had extra curricular and fun classes. Including biology dissecting, chemistry, essay writing, sewing, woodcraft, etc, etc, etc, etc.......whew! You name it, we (almost) had it. Whatever skill a parent had and wanted to teach. We had a lot of professionals or experts lend their knowledge to everyone. Those nurse moms wanted to dissect EVERYTHING! Ugh.
The one problem in Homeschool is that it is possible to OVER socialize sometimes and the mom has to pull back to STAY home to study. PE at the Y, Sign Language classes at the Institute of Deaf and Blind, every Field trip imaginable including a weekend long trip to Dauphin Island science center which included going out on fishing boats, AND a fun campfire on the beach and Alabama football on T.V. If we were learning it, we could find a field trip to cover it if we wanted.
Also a week long annual political camp in Montgomery (forgot the name of it) where they were able to create and pass a bill though the State congress (and eventually work for campaigns if you wished). All this was conducted during normal public school hours and is very rare in public schools.
Add church functions, sports functions, homeschool holiday parties, etc, etc, etc.

Homeschooling has really busy schedules and we had to turn down many events and field trips.
At college and on their jobs, my kids talk about how "unsociable" the public schoolers are. They are "trained" by their peers not to excell, but "hide" and not stand out. Sullen and angry is the descriptive words most often used.
I can spot a homeschooling family a mile off and will ask them if I am correct. Outgoing, sociable, creative, bright children who socialize with ALL ages. The teens socialize as easily with each other and with younger ages, as well as adults.
My daughter was on the President's List and Dean's list in College and will finish her interpreters degree & my son currently works at Southern Co. (a full time adult IT job) while finishing his college (which they help pay for
Upon graduating from High School, my kids both shared how much they enjoyed Homeschooling over all the other choices. They have both shared since then how homeschool helped better prepare them for REAL Life, college and their careers.