Like you said, the complaints are always from the population for whom it is accurate. They're usually in denial and typically vastly overestimate how much muscle they're actually carrying underneath their visible layers of fat.
Check the reddit/bulkorcut subreddit for some examples. If i had to spitball, i'd say only 20-30% of people that post their actually have a good idea of where they are in terms of BMI and body fat. There is an alarming amount of people that i'd consider their physique in the top 1% to top .1% in the world that believe they are fat. And there is an extremely alarming amount of people who are likely 25%+ and even sometimes 30%+ body fat that think they're in a healthy BMI range and ask if they should bulk.
TL:DR - BMI is more accurate than not for people who need the scale. And if anything, it underestimates obesity for people who may not weight enough to be overweight, but still sit at 30%+ body fat.