"is likely to spur an ideological showdown over the size of government and the role of entitlement programs likeand ."
The people serving in Congress got their jobs because they are effective politicians. This means that they have the ability to appeal to powerful interest groups; there is no requirement that they have any background in, or adherence to, any political philosophy.
The debates over competing plans for Social Security, Medicare, and Medicaid are most obviously about the distribution of income between the wealthy and the less wealthy. Most Republican plans for Social Security would substantially reduce benefits for middle-income, and sometimes lower-income, retirees. Democratic plans tend to be more likely to increase taxes on the wealthy. This is most immediately a question of whether money should come out of the pockets of middle-income people or wealthy people.
In the case of Medicare and Medicaid, the most obvious issue is between those who would want to revamp the health care system in ways that give less money to the pharmaceutical industry, physicians, and insurers, thereby bringing per person costs in the United States more in line with costs in the rest of the world, and those who want to protect the income of these interest groups and instead save money by denying health care to people. There could also be more taxes on the wealthy to support the maintenance of quality health care for the aged and the poor.
It is inaccurate to describe this as an ideological issue or a debate over the role of government. Both paths involve large roles for government. In the case of the Republican path, the government must play an intrusive role in protecting the patent monopolies of the pharmaceutical industry which allows them to charge prices that can be hundreds or even thousands of times the competitive market price. The same applies to the medical device industry. The government also imposes extensive barriers that keep doctors' fees far above those of comparably trained physicians in other countries.
The debate is not over whether the government should play a large role in the health care sector. The debate is whether the government's efforts should be devoted to maintaining the incomes of health care providers or whether they should be devoted to providing health care to the public. The NYT badly misrepresents the issue in a way that strongly favors the Republican position by implying that this is an ideological argument over the size of government in the economy. It isn't.