I don't think you're going to save on healthcare with visas like this. You're still required to be fully covered with your own health insurance as one of the visa conditions.
This removes the possibility of you being a burden on their public health system, which is common amongst not just these new 'nomad' visas but also the working holiday visas that have been around for decades.
Maybe. I'm guessing health insurance is still going to be cheaper to buy in any country outside US. A lot of time it's not worth to get insurance at all and just pay out of your pocket (because you aren't ripped off by hospitals).
But going back to point - why would anyone get this permit when they can stay on tourist visa. I guess complacency is one reason.