A few comments...
It is easy to train the body that most of us see in the mirror like the shoulders, chest and arms, but those groups are more or less the smaller parts of the body and less leverages and propensity for size than the large amounts of weight, that say the legs, back and hips can support or develop to lift.
Getting the largest structures of the body trained (legs, hips, back) will powerize the entire structural system and only add to better lifts in all areas IMO. Most guys who can OH press a good bit have pretty good hip, leg and back strength to back it up and support it.
I also never really think in terms of squats, being lower body specifically, since I thing good sets of squats, train a good bit of the entire body or at least up the back, spine and erectors along with lung capacity and cardio work. Same with deadlifts. They absolutely train more than lower back.
Take your time. Add weight slowly and steadily and don't sacrifice form for weight. You may be surprised how much you can eventually do for good solid reps.
Most guys have a good bit of room in the leg hip and back area, because there is a lot more room to get stronger for one, and heavy leg hip and back work can be uncomfortable for many to push hard.