I run a couple of Chris King BB's on road frames - 68mm shells, no spacers. They are direct replacements for the 7800 Shimano units, using the same fitting tools - some tools engage better than others though.
In use them with 7800 (DA 10sp double) chainsets.
I have not tried the latest CK offerings.
I have experience of various other BB's on my fleet (all 68mm shells):
Phil Wood stainless cup (old), direct replacement for shimano 7800.
Phil Wood aluminium cup (current model) supplied with 3x2.5mm washers this came up narrower than shimano 7800 when directly fitted.
Hope - road version, these come up the same as shimano when the 2x1mm washers are fitted.
Hope 68/73 version, comes with 3x2.5mm spacers, to suit road cranks it will run offset with one 2.5mm spacer.
Shimano R60, same width as 7800 but needs a smaller fitting tool because of reduced diameter.
Shimano 9100, same width as 7800 bits but needs an even smaller fitting tool because of reduced diameter.
As an alternative to spacing the cup(s) from the frame, there are 24mm axle spacers available in a range of thicknesses, but make sure the center tube doesn't get crushed or stop the cups tightening in the frame if doing this.
With the narrower BB cups the hollowtech and Miche Pistard crank arms bottom out on the splines before tightening on the BB, and need spacers somewhere.
Some (fatter) center tubes do not get on with internal Di2 cabling in some frames. Stainless Phil Wood has this problem.
Hope are the ones that don't use a plastic spacer to adapt a 25mm bearing to fit the 24mm axle. Pistard cranks don't get on with old (brittle) plastic spacers.