I think for some time there will remain a group of people, mainly elderly, who will have no access to the internet because its use is completely alien to them. The government making access available won't change that as it is not how these people operate.
My parents are both in their 80's now and comfortable with using the Internet. I think the day when the elderly people who are not comfortable with using it have all died is really not far in the future, maybe ten years.
OK, there will probably always be a small subset of people who do not understand the Internet and there will have to be a service available to them to help them get the basic services they need, but this need not be hugely expensive.
However, there will always be a much larger subset of people who cannot afford an Internet connection and this will be a more expensive problem to solve.