I think they care about their kids, as far as those kids are required to fulfill JB's dreams of world domination. By copiously breeding unto the Nth generations.
I don't want to go too far down this rabbit hole. But. There was this influential fundie leader Doug Phillips and his organization Vision Forum. They were riding high in the circles in which the Duggars (and Seewalds) move. Michelle was awarded MOTY by the Vision Forum. Then, well, Doug was discovered to be a sexual predator (he got caught using their live-in teenaged nanny as a sex object), and all the wheels fell off and VF shut down.
However, while VF was riding high, Phllips and this weirdo named Geoff Botkin came out with this thing called the Two Hundred Year Plan. I think it was mainly Botkin but Phillips was on board. And IIRC JB may have been influenced by this idea.
I believe JB had - and may still cherish - some ambitions along those lines. IOW, that his many godly children will marry and multiply and continue to worship the correct Jesus down the generations, and by some year in the future they and all their like-minded people who also worship the correct Jesus, will rule the world. Apparently by sheer weight of numbers, or the power of wealth they've accumulated. Or something. I don't think the plan allows for anyone to leave the family or depart from the accepted religious doctrine. Or be infertile. It's truly a control freak's wet dream, that 200 year plan.
I think that back when Jill was courting or engaged, there was a TH on the show or an interview somewhere, in which Jill made a reference to the idea of the Duggars living as this big tribe, maybe on a compound? Anyway, it was IMO a slip of the tongue; that's not something they are supposed to talk about in public.
I think JB believes that he can create and maintain a family-centered business model that will support his kids, grandkids, and the next generations too. There's no need for the sons to get secular educations that would just warp their minds and soil their souls. And of course the role of women is to joyfully serve the men.
So, IMO JB and Michelle care about their kids. As their legacy. At this time they haven't lost any kids to outright rebellion, but as has been said here, that's probably due to the TLC money keeping things comfortable. I don't know if JB's still on board with Botkin officially but I believe he nurtures some version of that dominion idea in his heart.