An interesting case study is Hong Junsheng's thought process on "Double Weighted" which was a topic in his book. It's interesting to see how he was confused and tried to address it.
Here are some quotes:
When I first started learning the Wu Style Taijiquan from Mr. Liu Musan, he showed me hand-copied versions of various Taijiquan writings. In regard to the issue of "double heavy", Mr. Liu adopted the common view of a Horse Stance being double heavy. I believed this theory at the beginning but eventually came to doubt it. If it is true that the balanced position of a Horse Stance puts the center of gravity in the middle and causes double heaviness, then the solution would be too simple: make one foot light! Why then does the "Single Whip" of the Wu Style still keep the Horse Stance? After all a Horse Stance is more stable than any stance that uses one solid foot and one empty foot. Why is it considered an illness then?
In my early years, I studied Wu Style Taijiquan. There was no toss to the left or right, forward or backward of the center when moving the body. Chen Style Taijiquan is even more precise. I do not know when or who started to say the center moves to a certain place. Even worse, the book "Chen Style Taijiquan" wrote that "the center moves totally to one leg.
"double heavy" [...] refers to the application of weight on both the front hand and the front leg at the same time, and thus corrects the misconception that it refers to the application of weight on both legs, as in a Horse Stance.
He learned from his Wu Style teacher about the horse stance being double heavy. He believed it. But he was confused as to why the Wu Style he learned had the weight in the middle in Single Whip. He believed that even weight distribution is more stable than stances that skew the weight in favor of a leg. He even read a Chen Style Taijiquan book that mentioned moving the center totally to one leg.
But in Chen Style, Hong's method seems like an outlier because everyone else clearly shifts their center from one leg to another in varying degrees. Yet Hong is pretty adamant about having the center fixed in the middle with the exception of footwork. In Single Whip, most Chen practitioners skew their weight onto the left leg.
To me, "Double Heaviness" is about "Double Yang". How and where we apply Yin/Yang is open-ended. But applying it to the legs is valid because you can apply Yin/Yang to all sorts of things. In this case, the heavier leg is Yang, and the lighter leg is Yin.
But regardless of terminologies, weight placement is relevant in a martial context. If someone's weight is in the middle, sweeping any of their legs is enough to unbalance them. That's because their balance is dependent on both legs. And if they want to step, knee, kick, or sweep, they need to shift their weight onto a leg in order to pick up the other leg. An extra action is required.
If someone's weight is skewed, sweeping the weighted leg doesn't work too well because most of their weight is there; it's like trying to sweep a tree trunk. Sweeping the unweighted leg is not as effective since they don't need that leg to stand to begin with. And if they want to step, knee, kick, or sweep, they have an empty leg already available. They don't need an extra action.