In the United States, as in the rest of the world, older people often believe that younger people live in a world with less hardship and more luxury than they grew up in. Although people's memories of the past often stress the moral superiority of the times when, for example "people could leave their doors unlocked", they like to give a contrasting picture of physical hardship.
Having to walk to school is an example of a hardship that children once had to undergo. I was born in 1979, and the history of this phrase was already well established when I was a child. Even at that time, it was probably a cliche, since many adults had grown up in two car families, or had suburban school bus service. This was true in the early 80s, and even more true now. Barefoot adds another layer of deprivation to the meaning, although again, one that is not very likely. People's memory of the weather also seems to be tinted, with the past being a time of immoderate extremes, and so snow fall was always greater in the past.
This phrase is of course, an exageration, and in case there is any doubt about that, the concluding line makes it clear that it is being used with irony, as the speaker admits the impossiblity of the situation. After all, while walking uphill is possible, walking uphill both ways certainly is not.
Thus, although the expression is no longer used seriously to comment on the weakness of children today (and perhaps never was), it is used by older people, or as a caricature of older people, to make an ironic comment on how people misconstrue their own past.