So this
diagram from the study I linked earlier puts ICE pollution at 25% MFG, 75% Operation (40% / 160%, from bottom figure).
Battery production all-up as 12% (15% / 160%) of an ICE's footprint.
Assuming you get your energy from a clean source(hydro, solar, etc) instead of traditional natural gas power production as listed in the article you'd see EVs as 55% / 160% so about 1/3 less emissions than traditional vehicle.
I'd agree with you that it's not 5x, but that's certainly no small drop either. Add that with less maintenance, cheaper to operate and 100% torque from 0 RPM I see EVs getting pretty compelling within the next 10-15 years.