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I had bad shoulders for a long time but worked through it by doing rotator cuff rehab exercises with very light weights for a couple years (a 'drawing the pistol' exercise on my side, front and side dumbbell lifts with ~5 lbs, etc), then dumbbell presses, progressing from fairly light weight ~20 pound dumbbells) up to 75-80. Just getting back into lifting after close to a decade out, but the shoulders are rehabbed enough to tolerate barbell presses with fairly strict form up to 135 lbs.
I don't think that the human body is designed? evolved? whatever. to lift much weight over our shoulders.
Until you get to very heavy weights, I know no one who has done damge to themselves on a bench press, or bicep curls. OTOH, about 50% of my age cohort (the old farts) at the club have done rotator cuff damage to themselves with weights in the almost-sissy class -in my case 30# dumbbells.
When you get a weight above your shoulders, the balance issue gets dangerous, because there isn't enough supporting muscle groups around if the weight gets off ideal.
"Until you get to very heavy weights, I know no one who has done damge to themselves on a bench press, or bicep curls."
Where do you "workout", Planet Fitness (that's not meant as a compliment)? Are you one of those weenies who do curls in my squat rack? Either you don't have a clue or you didn't put much thought into your comment.
I have treated hundreds, if not thousands, of patients who have damaged their shoulders (typically AC joint) with bench press - particularly wide grip bench press - or their biceps - typically ruptures in middle to older age gents.
Your anecdotal report that YOU know no one that has injured themselves in such a manner is beyond surprising. It's flabbergasting to anyone who has spent significant time either training with weights or treating lifters. As advice, your anecdote is dangerous and negligent.
Your statement that old farts do damage to themselves by using 30# dumbbells is - pardon the pun - something only a dumbbell would say. You have said nothing about what exercise they were doing with the 30# dumbbells.
If they were doing some type of ballistic, eccentric work, rotator cuff damage would be expected. What's far more likely is that they had pre-existing chronic rotator cuff damage and whatever ill-advised exercise they were doing (or the manner in which they were doing it or programming it) was the straw that broke the camel's back.
RE overhead lifts and supporting muscles - that's one of the effing reasons to incorporate overhead lifts - to develop the supporting muscles (ie the rotator cuff) so that you don't tear them in your ADLs. Also, the same point can be made regarding supporting muscles when doing the bench press.
Now, as far as some practical advice regarding overhead lifting:
1. If you have issues (pain, etc), see an orthopedic MD (not a chiro) and then get a referral to a good physical therapist. Like Jablonski indicated, rehab can take a long time, but it can help a great deal.
2. If you're over 30/40 avoid ballistic/eccentric oriented overhead work, i.e. what CrossFit calls wall ball. At a certain age, the tendons of the shoulder no longer have the requisite elasticity to tolerate and recover from the damage.
3. Dumbbell presses allow you to rotate the wrist to positions not allowable by a barbell. This accommodation MAY alleviate your shoulder issues.
4. Unless you're competing in powerlifting, it's safer - for your shoulders, not your ego - to do bench presses with a shoulder width grip. It's also more sports specific (except for the aforementioned power lifters - and maybe discus).
5. Low reps/ high weight can be more forgiving in terms of recovery than high reps and many sets.