Body fat %-based health is a far stronger health indicator compared to BMI. As it stands today BMI underestimates obesity quite severely (in quote below).

The reason why body fat %-based measurements aren't used is because of cost (depending on the type of test ) and training. Common methods:

DEXA is the most precise, but expensive. BIA (such as Tanita scales) estimates have a tremendous margin of error. Skin-fold caliper tests can have a margin of error of 1-5%, at least, dependant on how hard the healthcare professional is pinching the skinfold sites. Lean individuals tend to have small margins of error, whereas larger individuals tend to have much larger margins of error.

It also is less effective against athletes, although I would argue that athletes are a negligible portion of the population.

From your source:

BMI generally overestimates adiposity on those with more lean body mass (e.g., athletes) and underestimates excess adiposity on those with less lean body mass. A study in June 2008 by Romero-Corral et al. examined 13,601 subjects from the United States' third National Health and Nutrition Examination Survey (NHANES III) and found that BMI-defined obesity (BMI > 30) was present in 21% of men and 31% of women. Using body fat percentages (BF%), however, BF%-defined obesity was found in 50% of men and 62% of women. While BMI-defined obesity showed high specificity (95% for men and 99% for women), BMI showed poor sensitivity (36% for men and 49% for women). Despite this undercounting of obesity by BMI, BMI values in the intermediate BMI range of 20–30 were found to be associated with a wide range of body fat percentages. For men with a BMI of 25, about 20% have a body fat percentage below 20% and about 10% have body fat percentage above 30%.[28]