First of all, I think you can do a lot better than that for OHP. It will take time and hard work. You are young. The bench press and OHP both seem to help each other out. For example, I focused only on bench presses for years without doing any OHP's. My bench was around 250 lbs. When I tried OHP's just to seem how much I could do, I recall it being around 170 lbs. That's not bad.
The other crazy thing that I noticed is with when I took an entire year off from bench presses and focused only on OHP's for that time. I did it because I had shoulder problems from benching, impingement. When I went back to try benching one day just to see how much I could do, with someone showing me how to improve my form, I got up 275 lbs. That's a 25 lb increase on my bench without doing any bench presses at all.
Some of the best strongman competitors don't do bench presses at all. Jesse Marunde, who got 2nd place in the worlds strongest man contest, for example used to laugh when people asked him how much he could bench because he never benched at all. He'd focus on OHP's like crazy and I think he might have done inclines or dips once in a while as assistance. He never told anyone too much detail on how his routines were like, but my bet is that they were simple and focused mainly on a variety of strongman events and/or lifts that were similar to what he might encounter in a contest. Anyways, look up pics of him and tell me if he's lagging in the chest department. He isn't.
Jesse used to tell people to squat more when they asked how to improve their bench press. Why do you think that is? Focus on the basics, simplicity, work hard, and results will follow. If you improve your squats, your push presses (or jerks) improve. If your push presses and jerks improve you get used to handling more weight overhead and your military press goes up. When your military press goes up so will your bench. Not that benching has anything to do with strongman. When asked one time on his forum, he said if he could guess he could probably bench around 700 if he tried and focused on it. I believe that. The world record raw bench press is around 700 lbs last time I knew.
One thing to note about the bench press is it doesn't bother your shoulders as much if you arc your back and keep your elbows tucked in slightly. That's good powerlifting form for competitive bench pressing. However, that doesn't mean it's healthy. The problem with that is when you arc your back like that it squishes the discs in your back, potentially causing disc herniation and eventual back problems and pain. Not a good trade off if you're interested in health.
So, I think to answer your question, although bench presses are a descent upper body pressing movement, I think you're right. You don't need them and there are surely other choices that aren't going to hurt your shoulders as much. OHP's and pushups are great. Also, if you ever really want to focus on horizontal presses, floor presses are a descent option; because of the limited range of motion they won't bother your shoulders.
Another option if you really like benching is just to do them, but know where your limits are and don't do them all the time. Take a break once in a while. It's up to you.
A few other notes about shoulder health is it's good to stretch your chest and lats well because they are internal rotators. Stretch them regularly so they don't pull to much. Your band pull aparts are probably good because they work the posterior deltoids which are exernal rotators. Also include some sort of external rotation exercise to work the infraspinatus.
I hope that helps.