Most folks have shockingly bad shoulder mobility/flexibility. If they worked on that as much as overhead pressing they'd have fewer problems.
This is true.
This is a subject I've thought a lot about, i.e. how the hell I screwed up my shoulder in the first place. The simplest answer is that I was on the wrong side of the stress/adaptation curve, i.e. I overloaded the joint such that it could not properly adapt. This through some combination of my programming (which at times was ambitious) and perhaps my form, with a dose of predisposing factors in there (I have somewhat unlevel hips, and as a consequence, unlevel shoulders, due to an inherent hip asymmetry).
When the going gets tough in exercises, there is a tendency to spontaneously compensate. Be this a 1 rep max or the last rep of a really hard set. This, in essence, is the danger of pushing too close to failure too often - suboptimal technique with heavy loads. Not because you're trying to use ****ty form to lift more weight, but because when the going gets tough, we all compensate a little.
Different exercises seem to have different degrees of wiggle room. A deadlift gone a little wrong can herniate a spinal disc, for example, so REALLY pushing aggressively on that lift has a tendency to explode some people's backs, particularly when you factor in predisposing factors. Whereas with something like a standing barbell curl and the loads you're capable of using in that lift, you pretty rarely see people wreck themselves, despite the frequency of crappy form in most gyms.
In my case, for whatever reason, I would notice that my left shoulder would kind of spazz out slightly occasionally with either top end weights or in pushing towards the end of a hard set of standing presses. It's hard to say exactly why this happened, I don't really think it was a mobility limitation per se, and I've always balanced my pushes with pulls (both rowing and chinning in at least equal volume). Hell, when I hurt my shoulder, I was even actively doing prehab work for my shoulder in the form of external rotations.
I'm not exactly sure how I'd use this information, however. On one hand, I could have attempted to moderate this by staying well away from failure or top end weights in the standing press. But how realistic is that in the pursuit of muscle/strength?
As such, and as per bodybuilding wisdom, I do think there comes a point where you have to look at exercise variations which allow you to train the joints/ROM/muscles in a given lift but that don't predispose you to problems when the going gets tough. From a muscle building perspective I never HAD to do standing barbell overhead presses, and seated presses literally never hurt my shoulders. Though, unfortunately, after hurting my shoulder with the standing variant, seated started to hurt, too.
Either way, the landmine press seems like a way to get some loading in a press variant without aggravating anything. Over time, more exercise options may open up, though I'm not sure I'd ever go beyond just seated overhead variants. Again, from a muscle building perspective, I'm not actually sure I NEED any overhead variant. Bench variations and additional side + rear delt work would probably cover me just fine.