It certainly does make a difference for infant mortality, which is alleviated with proper pre-natal and post-natal care: http://www-personal.umich.edu/~baileymj/Bailey_Goodman-Bacon... ("Improved access to medical care resulting from the desegregation of hospitals and the initiation and expansion of Medicaid significantly reduced infant mortality (Almond, Chay and Greenstone forthcoming; Goodman-Bacon 2013; Currie and Gruber 1996b). During the 1980s, acute neonatal care (neonatal intensive care units) helped less healthy infants survive. Improvements in access to medical care also reduced the mortality of older children, especially mortality due to common childhood diseases such as pneumonia and meningitis (11 percent of all childhood deaths). Medicaid coverage played an important role for poor children as it increased care for conditions easily treatable with antibiotics (Goodman- Bacon 2013, Currie and Gruber 1996a).").

Of the various public health concerns, infant mortality is one of those that are most directly addressable with medical care directed at the poor. Improved access to health services won't get a poor person to change his diet, but can be very effective in catching potential complications from pregnancy, treating infections in neonates, etc.

Also, the study you linked-to doesn't support the claim you're making ("Giving poor people access to health care does not improve physical health in any measurable way."). The study looked at just three physical health outcomes, after two years: blood pressure, cholesterol, and glycated hemoglobin (a measure related to blood sugar level control). These are lifestyle-related measures unlikely to be affected in merely two years.

Moreover, the study is inappropriately designed to test the conclusion you assert. First, it is based on giving a random sampling of people access to Medicaid ("In 2008, the state of Oregon drew names by lottery for its Medicaid program for low-income, uninsured adults."). But to address lifestyle diseases, you need to build a culture of intervention over a long period of time. You can't do this in just a few years targeting a few members of the community. Second, the study doesn't measure the impact of "access to health care" but rather access to Medicaid. The non-control group still received healthcare services, they just did so through the emergency room and by doctors' visits that went unpaid ("Medicaid decreased the probability of having an unpaid medical bill sent to a collection agency by 25 percent.").