I agree, sort of. Incline hurts my shoulder, flat Is a little better, and decline is pain free. I must do dips wrong because they hurt my shoulder too. I also do cable crossovers to work the chest.
Incline just never felt right for me. I used to do it in the '90s but, like you, I felt it in my shoulders. I suspect I was doing it wrong, probably splaying my elbows too much, which can cause shoulder impingement. And despite shoulder issues in the past, for whatever reason, doing parallel bar dips through the full range of motion with as much weight I could handle never affected me negatively.
The only time I ran into a bit of shoulder trouble was when, following some misguided books many years ago, I did vertical dips with my legs straight, supposedly emphasizing triceps, after first doing sets properly with knees bent and a forward tilt of the upper body for pecs. In fact, I'm pretty sure the latter is the only proper way to do parallel bar dips, as it hits pecs, triceps and delts nicely. Simplicity.
I never did cable crossovers, so I cannot comment on them. They never appealed to me, although I did flies (flys?) when I was younger and more caught up in variety and volume. I imagine they're pretty much the same thing. Given that we can do less volume as we get older, all else being equal, I prefer to stick to compounds.