There are many expressions where i dag also means 'the present era', like, I dag lever många par tillsammans utan att vara gifta ('Today many couples live together without being married'). It's rather that when we need to create a genitive of that, we can say av i dag, but we don't really like that, it sounds a bit old-fashioned, so we prefer to have a real -s genitive, and we can't form one from i dag, so we have to resort to dagens.
It is quite possible to say Ungdomen av i dag too, and of course saying Dagens ungdom will also make you sound old-fashioned, the whole expression will do that because of what it means, but we really like to have a genitive expression rather than an av construction.
The day's youth should be accepted, not only is it a more literal translation; it's a correct one.
The day = today in many English dialects and needs to be added as a translation here.