With local natural forest it's likely better to mostly just leave things alone.
Maybe put your effort into invasive species and weed control? That seems to be an issue in most places.
Even than sometime the local wildlife often adapts to introduced species. Right now the native honeyeater birds (Tui) are camped out in my backyard because there is a big flowering camellia tree there, and flowers are scarce in Autumn. They will hang around town for most of the winter feeding on the exotic trees that flower late winter / early spring. (Banksia / dogwood / cherry etc) . When the native species start to flower, they all head back up into the native forest to breed.
There are all sorts of reasons you may want to prune trees. Timber quality, access, ease of fruit picking, utility clearance, aesthetics etc. But generally the trees don't "need" it. It's more to suit OUR goals, and the art is to do the least harm to the tree. Lopping off some lower dying branches so they don't poke you in the eye is "access", and does little harm to the tree, possibly helps it if done right because the dead limb scars over faster, reducing the chances of rot.