Any alteration to riding position to reduce weight on the hands, transfers that weight (and therefore any associated discomfort) to the bum. The only way out of this catch-22 is to get fitter.
It does indeed transfer weight to the bum (unless you push the pedals harder, as you say), but (1) a large reduction in the (small) weight on the hands results in a proportionally smaller increase in the (large) weight on the bum, and (2) luckily, bums are pretty good at sitting!
Hands, on the other hand, are bad at bearing weight for long periods of time.
I have ridden all day on rough roads (including ~30 km on rock-strewn dirt roads) with 25 mm tyres, thin bar tape, and no gloves without hand problems, all because of the advice in that fitting guide.
Core strength is needed to support your cantilevered torso, but no amount of core strength can do that if your weight is not correctly balanced for the force with which you’re pushing the pedals, since there’s nothing to brace against. The bracing occurs against the forward pedal or the handlebar. This is why most cyclists fall forward if they remove their hands from the handlebar while riding gently in a low position, even if they can trivially support their cantilevered torso in that position.
