I agree with this wholeheartedly, but Illich called for the abolishment of social institutions altogether, not the improvement of them.
I'm quite suspicious of such calls for abolishing institutions rather than reforming them - whether from the right or left - because they are often just cloaked attempts to replace those institutions with their own preferred "purer" institutions. In the case of Illich, his preferred institutions would seem to be somewhat connected to his religious work as a priest.
> dislike the US health care system, and think we should move from private insurance to a single payer model
Single-payer/Medicare-for-all is just advocating for reform of the health care system by removal of the pathologies of private insurance, not the abolishment of shared risk pools altogether. It's an evolution of the current system, not the destruction of it.
You can criticize a system while still participating in it; just because you think there should be a better system doesn't mean that the current system is worse than nothing.
I dislike the US health care system, and think we should move from private insurance to a single payer model. However, I still currently have private insurance and participate in the system, because I need healthcare and my only choice right now is private insurance.