I think it's a historical error to conflate 2-step's popular success with the presence of bass.
First, remember there was a 2 year lag between the first 2-step tunes (stuff like Anthill Mob's "Set You Free", Dem 2's "Destiny", Amira's "My Desire", Dreem Teem's "The Theme") and chart crossover, during which period (I'm given to understand) it was pretty much strictly a London thing.
Second, at the point of 2-step's crossover we're talking a low-ebb in bassiness, the booming (or wah-wah, in the alternate) basslines of speed garage had by this point faded somewhat. Stuff like "Rewind", "Sweet Like Chocolate", "Bodygroove", "Crazy Love", "Flowers", "Do You Really Like It", "Moving Too Fast", "Girls Like This", "Why?" (i.e. the tunes that actually crossed over) all had very minimal bass presence, and are for the most part pretty trebly.
(And as much as I love heavy bass tunes in 2-step, there's something equally nice about that fragile helium bittersweet bliss you get on a lot of the really trebly tunes - not a hit, but Artful Dodger's remix of Valerie M's "Tingles 2000" is an ace example of this)
Then the return to bass (esp. led by The Wideboys) actually started 2-step's drop away from popular crossover. But the scene had had a chance to establish itself outside of London by virtue of that window of crossover success.
I would say that as a general rule the primary crossover audience for bass stuck with drum & bass during that period and then switched to dubstep subsequently.
I think funky's lack of crossover success is pretty easy to explain actually, esp. by comparing it to 2-step at its most populist: by virtue of its distinct beat matrix and speed, 2-step was instantly identifiable even at its most smoothed out and songful and unthreatening ("Sweet Like Chocolate" probably being at the extreme point), so it gets the double benefit of being both interesting and comforting.
Whereas funky, when it smoothes itself out, is left with tunes that are practically indistinguishable from vocal house (e.g. Perempay & Dee's "Time To Let Go"), but if it stays jarring then it's... jarring. Esp. in a context where the rest of the pop landscape is disinterested in syncopation. 2-step had the benefit of coinciding with R&B at its most wired and jittery, so its rhythmic strategy made a certain amount of sense even in a pop context. Audiences are less forgiving of syncopation now though; I suspect this held back "In The Morning" somewhat.