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> We allow suppliers to set prices however they want, but we mandate that insurance buy the thing, which essentially gives them no negotiating leverage.

As far as I can tell, it's the opposite -- insurers have the most negotiating power by virtue of aggregated information about pricing and services and a revenue stream that's subscription/toll rather than service.

Patients are limited in their power to negotiate with providers by some combination of largely inelastic demand (nothing's more valuable than life itself, full health or as close to it as possible is a close second) and a lack of expertise (how do you know what you can forgo or substitute?).

Providers are in a better position (and generally do somewhere from pretty well to very well), but often carry heavy investment costs and must continue to provide services to collect fees.

> mandate that insurance co profits are a fixed percentage of insurance premiums, so they don't have any incentive to negotiate low prices anyway

This is only true to the extent that (a) they've already maxed out that profit and (b) don't have any other constraints. When you're suddenly forced to community rate, have new customers who've probably been under/uninsured joining your pool, have new care requirements to meet, dealing with an absence of lifetime limits, and limited in your ability to raise premiums without approval among other things, there might well still be financial incentives to reduce costs.