There can be a great argument on what is more important, what should be trained more or more focus put towards. But there is no argument that in order for the human body to move and perform tasks we need both to certain degrees depending on what task is asked of the body. There is no black and white issue here, yes both are important and need to be trained and appreciated.
Mobility is the ability for a given joint or set of joints to move through a certain range of motion, when we perform a squat and cannot descend down into a low position that can be a mobility issue through many joints, and the lack of movement in the body to allow the position desired.
Stability is the ability of the muscles and tendons to provide a secure position for the joint/joints and create tension to perform a movement, resist a force or move with force.
We need both and we need to have both to perform proper movement, if we lack either on patterns of movement will get thrown off, this is when we experience pain, tightness, limited movement etc. And of course if there is not a balance between the two then the system will try to create one or the other from joints or areas of the body that should not move in that manner, we create compensation patterns.
If a system is too mobile (hypermobility) that means that there is excessive range of motion that joints can obtain. This is someone who is very flexible and loose. This can be an issue due to the excess range of motion that can be reached, many times the muscles/tendons cannot produce the adequate force to stabilize or control the movement thus leading to injury, especially ligaments due to the muscles being lengthened too long. There is nothing wrong with this. Some people are genetically hypermobile, some need it to perform their sport but if you are in this category you need to train your muscles and brain to understand tension and proper movement control when weighted or training.
The opposite is someone who is owning the stability game. They are super stable and rigid, these are people who are limited in range of motion when performing exercises they feel tight constantly, they won't be able to obtain a deep squat, and have limited overhead range, due to the rigidity. This can be obtained by the lack of movement variations throughout life or developed patterns that are learned. It is a learned pattern that the body creates to live life, the range of motion exists and to unlock it we have to let the brain know its there. This can be changed by performing some mobility work with proper posture and position of the body and resetting breathing patterns.
So which to train and when?
Both are equally important, we need both but many will get confused on which to train and when and what is more important. All of the back and forth is irrelevant. The ability to have function and movement always comes first, so mobility takes the first step. If you train stability on a system with little mobility you risk poor patterning and tissue damage. Now training mobility is not hard, it's part of an activations/movement prep phase of the warm up, it teaches the body and lets it understand what ranges of motion are available and multiple reps are key to learning new patterns. Mobility is not just static stretching or holding a stretch for a long period of time, there is no influence on the joint, just the muscle tissue. So stop wasting your time, dynamic movement should be used for mobility, the passing through the entire range of motion.
Then comes stability once you have access to the appropriate range of motion then comes the task of how to control it and produce force, and tension and utilize the positions (range of motion) you have. This is your strength training, sprints, conditioning etc, the ability to produce force with various movements.
The biggest thing here is not sacrificing range of motion to lift heavier weight, we have all seen it, someone can bodyweight squat deep but as soon as they put a heavy load on their system it becomes a half squat. This defeats the purpose and is pushing you back towards low mobility, yes it can be lost, both can. So when performing strength training make sure to use the full range of motion during the movements this keeps mobility integrity and does not compromise it.