I would have thought, if AIB are going to cancel the overdraft, that you could not then withdraw or go over whats in the account.
Of course the best way is to constantly know exactly what your balance is and not go overdrawn, .
Your wife was upset about this but actually it is her fault not the banks. They are cleaning up their way of doing business and part of this is taking away overdrafts. Overdrafts were never meant to be a long term way of managing a personal account but people who have had them for years think they are a right. In your wife's case the banks thinks that her income does not justify an overdraft. That is their right.
You could have sorted this by making it a joint account and then the bank may have granted an overdraft.
By allowing your wife to go into 'unauthorised' overdraft it was far better for her. Hopefully, now yes she should know how much is in her account and budget accordingly.
And if you think you'll get any better service from a different bank please tell us because they are all basically the same, with in my experinece the BofI and AIB being so bad that I have vowed never to deal with them.