Those are trivial concerns. Any law/rule/entitlement that we've ever created is based on context, historical interpretations, and relies on various human factors. Coming up with questions doesn't poke holes in the concept at all.

>exactly how much healthcare access is a right.

Its not that hard to make a distinction between cancer and catching a cold. Pretty much all laws take into account intent, credibility, reasonable justification and other vague ill-defined criteria.

>what if the required resources are not available in the same amount any more?

Based on available funds, the codified law would be either stricter or looser on the qualifying criteria.

>Are people's rights now violated,

Yes, they would be.

>and if so, by whom?

You should be able to sue the party given the task of dispensing healthcare and make your argument.

In an interconnected society, pretty much everything relies on resources provided by other people. Consider these parts of modern society: Transportation, Electricity, Water, Clean Air (through enforcement of laws controlling pollution), Internet access, access to food, education, etc. Why is it that only healthcare gets the same treatment as these other fundamental parts of modern society? And why does it only seem to be from the U.S. political spectrum? No one else seems to have a major problem with this.

No one is saying you need a free ride here, they're saying if you pay your taxes, you should have access the same as for education, or water, or public roads.