I believe it is the governments responsibility to provide literacy to all. We have implemented a nationwide education system that covers children from 6 to 18 with twelve full grades and a kindergarten. These schools teach children to read and literacy is part of the curriculum. If a child fails to learn it is partially the schools (and the governments) fault.
This answer seems straightforward. We have given the government the responsibility of educating our children. It is in fact illegal to withhold your children from school unless you can prove to the government that you are offering them a legitimate home education. Since literacy is generally considered to be achieved before children exit elementary school, we have entrusted the government with providing literacy to all.
It is the governments responsibility to make sure that everyone of their citizens is literate. The government runs on trying to provide and protect its people but how can they say that when some of their citizens cannot even perform the simplest act of reading. Now saying that it is also the duty to all those citizens to help the people out that may need some more help or to at least push these people to going to a school for the problem.
Yes, it should be the government's responsibility to ensure that schools are meeting educational goals, including providing literacy to elementary aged students. The government needs to set guidelines for school districts and should take the necessary steps to ensure that each school is providing students with a fair and equal education.