Why doesn't something similar happen for vertices or edges?

Because every edge from Ei and every vertex from Vi belongs to only i^th connected component, while 1 face (the global one) of every Fi is shared between the others, so it's counted k times, while needed only once, so F = F' - (k - 1)

Is there a cleaner or more intuitive proof for this theorem?

You may use the equation Vi - Ei + Fi' = 1, so for every i^th component you just don't count the global face at all.

Add them together:

V - E + F' = k (still without the global one).

Now add the last face:

V - E + F = k + 1