Yes, you can weightlift. Things don't happen in a vacuum with regards to weightlifting. Much like bench press helps improve your push-up by strengthening the related muscle groups, and the other way around, any compound movement will hit your major muscle groups and stabilizers. Weighted squats will help your push-up, for instance, because a weaker person would likely have a weaker core, causing a dip when they do push-ups, giving them back pain f they hold that position for any real length of time.
You could do overhead press with dumbells for a few weeks and see an improvement in push-up and bench press, even if they don't feel like the same movement.