Here's the thing:
You can buy K12 yourself and use it privately as a homeschooler. I have heard mostly good reviews about it when it is used that way. In a sense, it is no different than ABeka, BJUP, or any other publisher, except that most of its "materials" are computer based.
You can also enroll in a government-funded, home-based public school. In some states they are charter schools (I don't know if this is true in all states or not; it is so in California). Some of these programs will use computer/Internet-based instruction, some use textbooks; some provide/require K12, some provide others (at least one offers Calvert).
All of these schools have actual names, such as Connections Academy. K12 is just the name of the instructional materials; it is not the name of the school
So, K12 is NOT an on-line public school. Connections Academy is. If you enroll your children in Connections Academy (to use that as an example; it may be a different school where you live; in California, it's California Virtual Academy--CAVA) the curriculum is not "chosen for you;" it is required that all students use the same thing.
"Graduation" only occurs at the end of 12 years.
Children enrolled in such programs are considered public school students, not homeschooled students.
I almost never recommend government-funded programs like this. If people want to homeschool, I can help them with that, because I believe that private homeschooling is, in most cases, the best thing for children. Public-school-at-home does not have the same freedom that homeschooling does. And IMHO, it's a way for public schools to suck parents in and have control over their children in a way that it is not possible when the parents homeschool.
Thousands of homeschooled graduates are accepted at colleges, including major, Ivy League colleges, all the time with nothing except parent-generated transcripts and diplomas (along with ACT/SAT scores or portfolios, if required). Enrolling your children in a public-school-at-home does not mean your children will have an easier time entering college than if you did it all yourself.
Your dh should look at state laws (especially the ones in the state where you live). You may live in one of the several states that are considered "green" because there is no oversight by the state in any way regarding homeschooled students. A few states require homeschooled children to be evaluated by a third party; several require standardized testing annually/semi-annually with results submitted to the school district or whoever, but the state doesn't usually give written acknowledgement that the children are where they "should" be. Most states don't required standardized test scores to be submitted to anyone, and those that do don't put anything in writing as far as approval or acknowledgement.
There are many distance-learning schools that require testing and whatnot. If you want to homeschool, I would recommend one of those (and keep my fingers crossed that your dh will learn to have faith in you and feel less need to depend on outsiders to determine how your children are being educated) over a public-school-at-home any time. I would also strongly recommend that y'all attend a homeschool convention next year, one with good workshops, so that both of you can be better informed regarding homeschooling.