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The great thing about the "lower body" powerlifts is that they are complex movements which train your entire body, not just legs.
I am talking about squats (barbell squats especially) and deadlifts. Here's 8 Reasons to Do Squat Exercises. Did I forget to mention that squats are hard? With weights, they are f-ing miserable. Body-weight squats are good fitness calis, but the weights are a bitch.
Lunges and step-ups are more purely leg focused, and in the calisthenic category even when done with dumbells. And the stair machines are great stressors, but mostly cardio and endurance.
I have found that doing five tough sets of barbell squats and 5 of deads (different days), plus, on different days, a set of goblet squats, of dumbell lunges, and a set of dumbell high step-ups, have improved my energy and ease of movement remarkably. Serious lifters would do those things twice weekly, but that's not me.
I can't credit at all to those, though, because I do other sorts of workouts each week too. Still, when you see progress with the weights you know it's all working.
Always been curious, what exactly do, for example, goblet squats do for you that barbell squats do not? Obviously don't contribute to strength improvement as much because you can't hold in your hands as much weight as you can on your back. Pretty much same range of motion, so it can't be helping 'mobility'. Three sets of five heavy barbell squats gets my heart pumping pretty good so I don't see the cardio advantage (and I would use something like a rowing machine for cardio in any case).
The purpose of the goblet squat is to teach proper form or patterning.
Most untrained lifters tend to back squat by bending at the waist, doing something that more closely resembles a good morning exercise than a squat. This is particularly true for low bar back squats.
Unless you intend to compete as a powerlifter, there are several reasons why this is not optimal, e.g. it's not ideal for the spine and it's not optimal for transfer to other sports or ADLs.
If you lift heavy and lift with poor form, sooner or later you will screw yourself up. The first rule of training is to not screw yourself up beyond what your body is capable of recovering from before the next training session.
Strength is not specific, it transfers directly to all aspects of human movement, making back squats an important, even vital, part of any training regimen. Done correctly, of course, as you point out.