Read the primary source rather than the write-up and the two do not agree, so here is what is actually in it.
The mechanism is slowed transit plus reduced intake, and that combination determines which interventions work. Less food means less bulk, slower motility means more water reabsorbed, and the result is a smaller, drier, slower stool. Insoluble fiber adds bulk to a system that is already moving slowly, which is why it frequently makes things worse. Soluble fiber holds water and is the better first choice, and an osmotic agent works with the mechanism rather than against it.
Where I think it is weakest: the subgroup findings are the part I trust least — with enough subgroups something is always significant, and these were not all pre-registered.
What would genuinely help is knowing what the mechanism is, because if it is transit rather than water then fiber is the wrong lever. If the honest answer is that nobody knows, that is a useful answer and I would rather have it.
Note on sourcing:
Figures above are from the primary publication rather than the press summary. If a number here disagrees with one you have, post yours and we will work out which of us is reading a secondary source.