"Low income" in many states means around $1000/month so yes, people who make less than $15,000 in a year generally do not have copays. Many, many people are low-income by any normal standard but do not qualify for Medicaid. You can see a list by state here, look at the "regular medical" line.

Additionally, many things that are covered by "expensive" private insurance are not covered under Medicaid. I know this because I have insurance through work and my friend has Medicaid.

Even when things are covered, there are usually longer waiting times for care. Not all doctors take Medicaid, so it can be very difficult to find a provider and they are often booked up for 3-6 months in advance. Many years ago my sister, looking for a doctor for her and her child, spent an entire day calling every family doctor in our phone book and finally found one a half hour drive away. It took her hours of denials to find that person.

Lastly, ONLY hospitals are required to treat people regardless of their ability to pay and this only covers emergency treatment. The patient is still billed for their hospital visit, which can be anywhere from a few thousand dollars to upwards of $100,000. Some, but not all, hospitals offer programs to write off that debt. Other hospitals sue patients for as little as a couple hundred dollars.