Let's clear one thing up first: aside from a few issues like joint stress, it's not weight in general that's the problem; it's body fat. If you're a power athlete with significant muscle mass, you're going to be able to carry a lot of weight and still be perfectly healthy as long as it's lean weight. Dwayne "The Rock" Johnson shouldn't worry about his weight, despite having a BMI that's technically in the obese category. If you have a high weight for your height and you don't look like a professional wrestler, you should definitely be concerned about your weight.
Aside from the problems caused directly by excess body fat, carrying around excess body fat also makes it far more difficult and far riskier to exercise, which means all your other health indicators are going to go down south. Even mental health is pretty strongly influenced by exercise. Also, excess body fat is a far more difficult problem to solve.
I would say this is putting the cart before the horse. It's not excess body fat that is the source of health problems. It's the lifestyle (diet and movement) that leads to excess body fat (as one of the many symptoms of growing health problems) from choices.
In other words, baldness, obesity, sluggishness, headaches, etc... are not the source of health problems - they are the symptoms.
(Though one imagines that if one is not willing to change one's diet or exercise habits to lose weight, the odds of them willing to do it to lower blood pressure or other such indicators seems low too.)
It's true weight is very often a negative health indicator because of the toxic nature of the diet that results in weight gain. However, losing weight is not always a sign of being more healthy.
Shutting down your liver, clogging your arteries, and generally filling your soft tissue and fat storage with toxins is not healthy even if you appear to be losing weight.