Actually, we are both somewhat confused. Your post above has helped me see where I had some things wrong. I was forgetting that not everybody is eligible to get Medicare, even if they are over 65 or have been on disability income for over 2 years. As you say, you have to have made a certain amount of payment into the system during your working career through payroll taxes. My neighbor, for instance has only worked under the table doing house cleaning. She gets no regular social security or SSDI. Now in her 70s, her income comes from SSI. She is not eligible for Medicare. She gets Medicaid. I was thinking that everyone over 65 gets Medicare, but, as you say, that is not true. You must have paid into it to a sufficient extent. If you have worked enough to qualify for Social Security or SSDI, then you qualify for Medicare, with 2 year waiting period in the case of disability. (Some people qualify on the basis of a spouse's work and payment history.)

Now here is a slight confusion on your part, if I'm understanding what I think I do. SSI does not stand for "Supplemental State Income." It stands for "Supplemental Security Income," and it is basically a welfare program run by the federal government. Here's a source to back that up: https://www.ssa.gov/ssi/

Some states, especially those where the cost of living is high, provide additional money to those of its citizens who are on SSI. Many states arrange this by handing those funds over to the federal government to put into the federal SSI check. Some states pay the amount separately. This extra money is called the State Supplemental Benefit. Here is a link explaining:

What Are State Supplemental Benefits for SSI Disability? | Disability Secrets

Some people do get both a Social Security check (either regular SS, or SSDI) and a SSI check. These are people whose Social Security check, or SSDI check, is very small, meaning below what the federal government pays in SSI. People like that get Social Security (reg or SSDI) plus SSI (to add up to what SSI pays those who only get SSI) plus whatever their state gives to people on SSI (the Supplemental State Benefit.)

These people (in the paragraph above) would then get both Medicare and Medicaid. They are called "dual eligibles." Here is a source:

Dual Eligibility ? Medicare and Medicaid | The Medicare & Medicaid Center

What is important to remember is that it is the federal government who decides if you qualify for basic SSI - not the state government. The state only decides if they want to add something to the SSI.

One of the things the state can add is to pay the part B premium for low income people on Medicare. It can also pay that for people who get no SSI. I get that. In my state, that benefit is called a form of Medicaid. The state Medicaid people administer that benefit. That is true in most states, but maybe not in every state.

There are people who do get total Medicaid piggybacked on to Medicare, as I said further up . . . and those are the "dual eligibles." There are plenty of doctors who accept Medicare, but don't accept Medicaid. On the other hand, it would be very odd for a doctor to accept Medicaid, but refuse to accept Medicare, since Medicare pays the docs better than Medicaid. But it possibly could happen. A doctor certainly can accept both Medicare and Medicaid. As you say, doctors who accept Medicaid often tend to be the bottom of the barrel. As you say, few doctors can afford to not accept Medicare. Those who do reject it include doctors who cater to the very, very rich.

When people have both full Medicare and full Medicaid, Medicare is the primary insurer and Medicaid is the secondary insurer. Medicaid would pick up co-pays, for instance.

It is true that people who only have Medicaid do sometimes get care that is not the best. Sometimes they do have trouble finding doctors. State certainly can differ in how well they compensate doctors who accept Medicaid.

Like you, I get my Medicare Part B premium paid for by my state. As with you, I do not have a Medicaid card. I only have my Medicare card. A doctor seeing me would actually have no way of knowing that I get a state subsidy to pay the Part B premium and no reason to care that I do. It does not affect what he or she gets paid. In my state, I get letters saying that Income Support in my state is paying my Part B premium for me as a form of limited Medicaid assistance. Here is a link that shows that most (and possibly all) states call that assistance "Medicaid," even though it doesn't get you a Medicaid card:

Getting help to pay Medicare Part A and/or Part B premiums (the Medicare Savings Programs)

Excerpt:

You automatically qualify for Extra Help if you have Medicare and one of these:

Full Medicaid coverage

You get help from your state Medicaid program paying your Part B premiums (in a Medicare Savings Program)

You get Supplemental Security Income (SSI) benefits

("Extra Help" is a federal program for low income people that helps pay for prescriptions. It is also called "Medicare Low Income Subsidy for Prescriptions," and I get it. Thanks to that program, I pay no more than two dollars and change for a month's worth of any prescription.)

Like you, I was very concerned about having insurance that is portable across state lines. As with you, the fact that my state pays my Part B premium does not affect my Medicare being portable. As you say, full Medicaid does not typically cross state lines. (Actually, I think in case of an unforeseeable emergency, it can, but I don't really know.)

I think I've got this stuff straight, but, if anything is challengable, I would be glad to have that pointed out. Thanks for your post above correcting something I had got wrong.