It does not take a lot of strength to throw a 5 oz. object - period. That is not to say that people (or pitchers, for that matter) should not work out. Sure, Lincecum is strong - but what kind of strong? What is is max deadlift? What does he bench? I bet not so much - nor does he care. It’s not his objective. Always go back to what your goal(s) is - what are you trying to accomplish…
I am a big advocate of working out - but to what sacrifice? And how to best work out? What are the goals? If it is to be a better pitcher, well, if I am already a fully developed athletic male (which I am), then I may not do much other than to pitch. I would want to spend my time perfecting my craft (a skill activity). But, I am interested in long term development and improvement of my physique and fitness abilities so I choose to work out regularly.
That being said, I think youth athletes should also exercise with the goal of developing long term fitness. So my suggestion is to learn how to train for the big picture - not just for baseball. To get stronger has a lot of different meanings and lifting weights is a very small part of the puzzle (maximal strength). Athletes should place equal focus on things like explosive strength, starting strength, acceleration strength, strength endurance, high speed conditioning, reactive strength, speed strength, and flexibility training. Now, as a young athlete playing a sport(s), you need to spend time practicing your skill(s) for your sport - not spending all your time in the gym, so workouts need to be short, yet intense.
Ultimately, I believe that once a pitcher knows how to control his body movements (functionally strong), he is strong enough to pitch. From there, it is perfecting mechanics and target practice. There is no “specific” muscle to train to make one throw faster. It’s all about speed of movement and elastic energy, not muscular contraction. The arm is just the delivery device of the accumulation of strored elastic energy from the larger muscles of the legs, the hips, the abdominals and the trunk. This stored energy is transferred from one part of the body and handed off to the other. When mechanics are efficient then the arm will get most of that stored energy and that stored energy will convert to arm speed from trunk rotation. Sure, there are studies that say this and there are studies that say that (both for and against long toss and weighted balls and strength training, etc.) but most of these studies are seriously flawed; you just can’t make a study out of young athletes due to the fact that the overriding factor in their progress is due to natural growth and development. The only real studies pertaining to sports should be applied only to high level, fully developed athletes (collegiate and/or professional).
Look, my kid is 13 years old - 14 in a few months. He has barely touched a weight and has never long tossed and never picked up a weighted ball. This year he started working out 3-5 times a week in the off-season by doing all kinds of varied workouts; speed work, flexibility work, explosive movements, etc. He can do a split and he can do handstand push-ups, one arm push-ups, and one legged squats. He can rip off pull-ups like crazy and he can do 100 burpees in record time. His workouts are fast and furious - 20-30 minutes (never more than 40). But mostly, he pitches. In the off-season we start with once a week indoors - 50 pitches or so. After a month or so, we go to twice a week, gradually increasing the number of pitches. In February he goes to 3x per week and will be at 80-90 pitches per session - and it’s all about mechanics (videotaping) and target practice.
All this being said, my kid, for the last two seasons was 5’ 2" and barely 90 lbs - as a 12 year old he was barely hitting 60 mph, yet last season he was near 70 (before ever working out). He is now almost 5’ 7" and 115 lbs - still skinny (just now has a deeper voice and is the second tallest member of the family). My point is that he doesn’t care about packing on pounds or getting bigger and stronger - he just wants to be fit and agile and athletic. But mostly he focuses on what he has to do on the mound. He doesn’t concern himself with size or strength or how fast he throws - he knows all that will come - in time. It still comes down to making pitches; executing one at a time to the best of your ability.