The pruning is fine as the cuts appear to have been made cleanly. You can, for aesthetic reasons only, remove the dead stems when they have dried out enough to expose the trunk. I normally do this by hand under the theory that if they are so strongly attached you cannot remove them then they need to dry further.
Start at the bottom of the trunk and try and twist off the lowest dead stems. They should come off cleanly and expose more of the trunk. I stress again that you would only do this if you have time on your hands. There is no reason for plant health to do this.