Insurance companies do not want cheaper care.
In the US, insurance companies must spend 80% of premiums on care. So if you pay $1k/mo, they have to pay out at least $800/mo in care. (Not to you specifically, but averaged out across all subscribers.)
This is a cap on their potential profits. They always have to pay out 80% of premiums for care, so how do they make more money?
Well, imagine care is twice as expensive. Instead of paying $800, they have to pay $1600. That sounds worse, but, instead of $200/mo, they now $400/mo for themselves!
So, no, paradoxically, it is not in the interest of people paying for the treatment to save money. Quite the opposite.