Unfortunately, the nature of the stability of the shoulders and knees are much different. First, the knee is a much more stable structure than the shoulder. Also, at the shoulder, stability is enhanced by BOTH musculature and ligamentous structures, whereas the knees are more reliant on ligaments (ACL, MCL, PCL, LCL). Your best bet is just to focus on flexibility and balance between the extensors and flexors.
Interesting question. I have noticed in the past when I trained with more infrequent workouts and less volume I had much more trouble with the knees then I do now even though the total volume of everything (weights, sprints, gpp etc.) is a lot higher which really doesn’t make a lot of sense. I’d say there’s definitely a place for doing light feeder workouts and other activity in addition to your heavy weight training workouts to get blood into the muscles and to help repair and regenerate tendons, ligaments and other tissues. I had quadriceps tendonitis for 2 years in both knees that finally resolved when I began doing rope skipping. The ligaments, tendons and such need more volume and lower intensity to strengthen and regenerate than the muscles. If you’re only squatting once per week and doing nothing else there is a chance that your muscles could get really strong while your supporting structures actually weaken.