Yeah, healthcare in this country is a hot mess in a lot of ways. Something that could help push it in the direction of getting coverage is if you have any family history of things like uterine fibroids, or gynecological cancers. It’s a pretty straightforward thing on the paperwork end of things if cancer prophylaxis is on the list of reasons.
Another thing you could consider in this capitalist hellscape is signing up for a plan off the ACA that has a deductible similar to or less than your savings. That way you would wipe out the deductible immediately, have access to more providers, and have some semblance of coverage for the rest of the year.