Pitching is a movement, pitching hard is the most explosive movement in sports, thats a whole body movement. Strength development is always good. Whole body strength should be addressed. First however I always recommend a movement screen done by a qualified therapist/trainer. In particular after an injury. I prefer a physical therapist with a working knowledge of the sport over a doctor for this.
Lower body work is usually things like squats, dead lifts, single leg variations, lunges ect.
Vertical jumping doesn’t have a huge place other than general power development, but, lateral jumping does.
Core work should be rotational in nature. Doing crunches ect. or “bro” work outs for six pack abs don’t have much carry over. The mid section of the body in pitching stabilizes but also has a big rotational factor. Pallof presses (stabilization) russian twists (stabilization & rotation), medicine ball work (rotation) are all great.
Upper body work, in my opinion, should have a 2 to 1 backside to frontside mix. So, more work on the posterior or back side of the upper body. The reason is about 70% of injuries occur to pitchers at or after ball release. Slowing the arm down is where a lot of guys have issues. So, for example, doing a set of Zottman Curls for the biceps…then do two sets of triceps.
Triceps, Zottman Curls (better than normal curls), lat pull downs, pull ups, dumb bell bench press, push ups with scapular pinch, heavy bag work (punching bag), rows ect. A lot of different stuff that can be done.
So, IMHO, first get a movement screen done and address any existing limitations.
Free weights over machines. Machines were designed for body building. Lifting on a program that is based on a body building model (3 sets of 12, or sets of 20 etc) is not ideal. You want to lift to get powerful. Strength over size…and no, those two are not always related. Quality of lift over quantity. Things like squat and dead lift are going to be a little perilous for you at first with your back and I recommend spending the money to work with a qualified trainer to make sure your technique is good. The other reason free weights are better is they are movement based. When you do a squat or a dead lift for example you are putting the body through a movement, the hips are engaged, the core is engaged, the body has to stabilize and balance itself through the movement while moving weight…if you are sitting at a machine designed to isolate a muscle there is no movement going on. There is a lot more cross benefit to lifting free weights.
Forearms too. They are often forgotten but the flexor pronator mass stabilizes the elbow through the throwing motion.
Lifting can have great effects for some and not much for others in terms of velo. A lot of variables there in terms of existing strength, mechanics, ect. However, in terms of staying healthy and recovery it is always better to be strong than weak.
Arm care first, throwing second then lifting. That has been the approach I use.