The POA has the power granted to him under the power of attorney document. It may or may not give him access to the bank account. The POA is supposed to act in the best interest of the person for whom he is acting that may mean he could change names on the accounts if it is in your's mother's best interest.
If a guardian is appointed he or she will have to inventory all of your mother's assets including the assets in her box. If you think the inventory is not complete exceptions can be filed. The guardian is supervised the court and needs to get court permission to act.
