It's important to know how these are structured. K12.com and Connections are two competing national for-profit curriculum companies. You can access their curriculum in basically 3 ways, at least from what I know of k12.com.
(1) You can purchase their (or anyone else's) courses separately, with no teacher and no credit, for a modest price (maybe $1k/yr total for all courses, though there are varying pricing options), and basically be an independent homeschooler,
(2) You can go in their private school (expensive, maybe $5k/yr), and have online teachers and get credit for the courses,
(3) Public Virtual Charter School. It's free, and is essentially equivalent to (2) except that the online school itself is not run by k12.com, but instead is run and staffed by a (usually not-for-profit) state or district based public charter school. It is subject to state and local rules, and the same bureaucracy/politics that B&M schools can have.
Also K-8 is somewhat like homeschooling, while in high school the teaching is more done by online (human) teachers.
The national for-profit curriculum companies are definitely trying to get into every state and region, but where they are absent it is due to local obstacles. Also many other drawbacks of the Public Virtual Charter Schools stem from state/local rules.
Our experience has been that we can go at our own pace. In two years, our son has covered 6 grades in maths, and 3 grades in other subjects, though in a couple of cases he was blocked from moving to the next course. Also, there are various strings attached (that an independent homeschooler doesn't have) like logging "attendance", class connects (teacher led online sessions which you may have to waste time watching (live or as a recording) while they duplicate the material already covered in online courses), work samples (submit proof that your actually doing the work and not just faking it, to satisfy the school and the state), and they may make you do stuff on Study Island, etcetera. The advantage, compared to being an independent homeschooler, is that you have more structure, and you officially get credit for courses taken, but the disadvantage, compared to being an independent homeschooler, is that you give up some freedom.
You need to check with your local Public Virtual Charter School to check how they treat acceleration and gifted students, as there is much variation between regions. (And be aware that high school is less flexible.) Our school has allowed most but not all acceleration.
As for daily or weekly schedule, you can pretty much do whatever you want, any time, any day, provided you get the work done. This allows us to travel, if we want, and go to places that aren't busy because most kids are at school. But again, beware that some schools may not allow such flexibility.
Check with your local Public Virtual Charter School (if you have one).
It has worked for us, mostly due to the ability to accelerate, despite other drawbacks.
