The biggest assumption is that we want to control our children, are fearful and overprotective, and don't care that we'll raise parochial adults:
Appreciation of ethnic and cultural diversity, and thoughtful exploration of ideas of all kinds, are beautiful, necessary things. And there is absolutely no reason to think that a traditional school setting is necessary to accomplish any of this.

Anyway, our first priority is academic excellence. Followed swiftly by enhanced family life; and yes, the chance to connect deeply with our children, have a close hand in their character formation, and ultimately raise countercultural people.
Is it lonely and difficult to be countercultural? Yes, at times. And...? My children will survive - thrive, I'd say - if they are not "culturally literate" in the newest television shows. Far worse for them to be illiterate in true Culture, something that, despite a parent's best efforts, happens all too often when such a substantial portion of time, energy and focus is spent in typical school life. And far, far worse to be unduly comfortable in this culture of individualistic, consumerist, self-centered atheism.
Unless your child attends the very rare school with a countercultural ethos (and an avid champion like Clare) chance are excellent that you've got the rule of lowest common denominator, or at best "average," in every way - virtue formation, pace of learning, group-think, rule of the mighty and the beautiful. This demands compromise of self to survive and "succeed." It promotes the suppression of free thought and true development of individuals. No thank you.
Or think about the lack of love-based supervision and intervention guiding these interactions and behavior choices at tender ages, choices which over time amount to habits, which amount either to established virtues or vices - the true stuff of happiness or unhappiness, success or failure.
Then consider that opportunities to meet peers and socialize, in smaller, more natural, less exhausting settings - settings not intended, by the way, for focus on learning history or math or reading - are abundant.
And also consider that socialization inside the family unit, with diverse ages and needs, based on love and loyalty and the inability to just walk away and go home with your toys, is THE best and truest kind there is.
It's a big, beautiful, complicated, diverse world out there. And we want a piece of it - a much bigger piece than we believe our kids are going to get sitting at a desk in a typical school. We realize this is an ideal; and that there are many circumstances that could make it untenable for a year, or more, or maybe forever. But we can't deny that in our best estimation, it is the ideal. And we pray we're up for the challenge.

P.S. I'll say this just once: This is not a criticism of others' decisions, or our own whatever they end up being. It's an exploration of an amazing opportunity we're passionate about; and that, if we pull it off, will comprise a huge part of our world, of which you all are a beautiful part.