I'm not sure I necessarily agree with the term 'modern classic' for Emily as, like you say yourself, there are some pretty famous historical examples. Sure, it might have dipped in popularity over time, before becoming very popular again in the 90s/00s, but there are a fair few old ladies around called Emily too. It's not like it disappeared completely.
Names naturally dip in and out of usage but a classic is a classic. I don't think they need to be defined by era. I mean, Sarah really isn't very popular these days, but it obviously still has classic status.
For girls I'd say the following are all obvious classics, despite all being more or less popular at different points in time:-
Elizabeth, Victoria, Anna, Hannah, Emma, Emily, Sarah, Sophie, Lucy, Catherine, Grace, Rose, Rebecca, Rachel, Olivia, Alice, Eleanor
There are no doubt some obvious ones I'm missing but those names are never 'out' of fashion even when they're experiencing a less popular spell as they're always around to some degree.
Boys equivalents would probably be James, Jack, Thomas, Samuel, Daniel, William, George, Edward, Charles, Joseph etc. They're always in use throughout the generations.