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shoulder press is a shoulder press
A little personal anecdote. I've developed a rotator cuff problem in recent years. Weirdly shoulder press didn't hurt it. Turns out that I had adapted my form slightly to compensate and it was excruciating to do correct form. I compounded the problem by doing this.
I'm not exactly inexperienced with lifting weights. I used to teach Olympic lifts when I was a PT (20 years ago now). I'm very out of date and have forgotten a lot (not to mention that opinions have changed) but I know the basics and importance of form.
I have found it extremely useful to be micromanaged on form every now and again to help me get back on track. A shoulder press can quickly stop being a shoulder press if you're not careful.
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But that level or correction could easily come from a session where a coach is present, never mind 3 hours of PT a week.
I'd probably also point out that something insane like 75% of the population have dodgy or damaged rotator cuffs to some extent, lol. I stand by the fact that autonomy is more important, and by all accounts you've identified and corrected the problem yourself!
But christ, a shoulder press is a shoulder press. I feel like micromanaging each exercise is part of the culture of promoting the need for a lot of PT time.