I'll leave that as an exercise to the reader; I'm not going to wade through all the legislation that created CMS.

Then I think you'll be sorely disappointed in the outcome of this, the EO you wanted. If the legislation that created CMS doesn't give them the authority to require medical providers to publicly disclose their prices, then they're not going to be able to do it. And an EO can't change that.

Give the text of the proposed rule you linked above, it doesn't look like they have that kind of power. They hung the power to require the disclosure of insurer's agreed reimbursement rates on a pretty thin reed, and one that wouldn't reach to actual medical providers. If they had a statutory grant of authority that was broad enough to force providers to disclose their prices, they absolutely would have pointed to that as part of their discussion of the legal grounds for the rule. They didn't, so it seems pretty likely that in "all the legislation that created CMS" they were never given this kind of authority.

As I mentioned upthread, if you're looking for the federal government to start doing more than it was doing before, you've come to the wrong Administration. The conservative legal project is to put a harness on federal administrative power, and DOGE and the Administration are going to slash agencies' resources to the bone. This is not an era when the federal government is going to meaningfully take on new regulatory efforts.