Why do climbers tend to have are forward hanging posture, if climbing strengthens the back muscles?

This is something that confuses me and apparently even some (non-climbing) physiotherapist. Mine, claims that my climbing training has lead to my shoulders being pulled forward (it looks like he's right) and I should hence train my shoulder/back muscles. He told me to NOT train the chest (e.g. pushups).

However, I've read that climbers do pushups as a countermeasure because climbing mostly trains the back muscles. Which one is true?

It seems logical to me, that climbing is training the back and hence pushups are a good counter training. What confuses me, apart from my doc, is that climbers tend to have a forward leaning posture. Shouldn't it be the other way round, as shoulder training shortens the muscles and pulls the shoulders back?