The obligatory part is the good part. In the Netherlands it is obligatory and costs are lower than in the US.
I agree with you on the cost comparison side - I've lived in both places and find the systems quite similar in many respects outside of costs.
I think one key difference between the Swiss / American model and that of other western EU countries is the cost of billing and processing. In the US a full third of the cost of care is this part of the system, something that doesn't need to happen when there is a single universal payer for all care.
Obligatory insurance destroys competition (and insurance companies are lobbying hard to support it), because obligatory customer is a perfect customer — you can be as bad as you wish to them, and they still won’t leave you. With multi-payer insurance, they can still leave to other company (but it still enables bad behavior in the insurance industry in general). With single-payer, no such luck, you are at the complete mercy of the single-paying entity.
In a single-payer system, there can still be many healthcare providers, and the single entity doing the paying is incentivized to push aggressively for low prices from providers, because that keeps their own overhead low. Likewise, they’re incentivized to keep their premiums low, because when you’re the government, if people think you’re screwing up their healthcare they vote you out of office.
As an example of this in the US, take the Medicare system, which can negotiation super aggressively on pricing from healthcare providers because a provider who lands a contract for Medicare knows they have a massive pool of humans to provide care (and thus receive payment) for/from.
I called out the things that keep single-payer systems in check in my original comment, which you’ve replied to: pressure from constituents keeps premiums low (in the same way that the government is pressured to keep taxes low by their electorate), and they’re incentivized to negotiate low rates with providers because it improves their ability to keep premiums low without having them be lower than expenses.
That said: the US’s current privatized system is sloppy and corrupt in a multitude of ways, because their incentive structure includes turning a profit, and they can’t negotiate costs as hard as a single-payer system can, because their pool of customers is smaller. So we’ve got the worst combination of traits.