Phil's doesn't have a strong, offensive smell. It's just a distinct strong odor that is detected when you're close to it. The silicone base is probably what makes it difficult to remove from your skin and the smell lingers. The factory Shimano stuff has a similar odor.

It must have been some other brand if it was that bad. I once cracked open a Specialized branded BB and it stunk so badly I had to hold my breath. It reminded me of the stench of a car salvage/junkyard. All that drained fluid mixed into the dirt, rainwater, cooking in the sun for years. It's foul.
Gear lubes use sulfur compounds as an extreme wear additive, they stink! Anyone who has changed their transmission and/or differential lube will know. Maybe some grease brands use it too.