Maybe you need an extra day in between so that you can consistently hit 5 reps each time. If you are older, that is likely the case. Or perhaps you just need to rest longer between sets.
As for the quote you posted, the reason it is true is because it's that whole stimulus response cycle. The stimulus is more weight. More volume and frequency at the same level isn't enough of a stimulus to make you gain strength.
Female, 5'3", 49, Starting weight: 163lbs. Current weight: 135 (more or less).
Starting squat: 45lbs. Current squat: 170 x 3. Current Deadlift: 220 x 3