While I am hardly an expert, I have suffered a good bit from joint inflammation, and have subsequently gone down a lot of rabbit holes in a quest for relief. I'm not convinced that caffeine, in and of itself, is that problematic. For me I think it even has a mild analgesic effect. I think the problems come more tangentially. The delivery mechanism can get you. Sugar will absolutely ravage you in terms of inflammation, so if you're getting your caffeine via sugary coffee drinks, sodas, energy drinks etc., that could definitely be a problem. I tend to believe that the chemical sh!t storm that is artificially sweetened energy drinks and diet sodas has a similarly deleterious effect. Poor sleep will also rack you, so if you're jacking up your circadian rhythms by having coffee later in the day, that could also get you. But black coffee, or better yet, unsweetened coffee with healthy fats (MCT Oil, etc.) at the right times of day, I say game on. *It should probably be noted that I am a full blown caffeine addict, so it's possible that there is some confirmation bias in the research I've done, and of course everyone reacts differently to different stimuli.
All this said, inflammation and injury/mechanical deficiency can be two very different things, and shoulder mobility/stability is it's own rabbit hole. If you have some time, it might be worth looking up Range Of Motion fitness on Facebook. I have no affiliation whatsoever, but the guy who runs it has posted a ton of free videos, some of which I've found pretty helpful with my shoulder issues. He's kind of like a poor man's Kelly Starrett, but I can actually understand what he's trying to say. He's also based in Riverside, and does personalized mobility assessment and plans, though I haven't partaken in that yet.