Is the statement of the brother—who is a religious teacher—that it is sufficient after washing the hands to shake off the water from them and then wipe the head, correct for someone who suffers from waswas (obsessive doubts)?

If what your brother advised you to do was to wipe the head with the wetness remaining on your hands without taking new water for the head, then this is contrary to the opinion of the majority of scholars. However, it is the Maliki school of thought, and it is a sound and recognized madhhab (definition). You may follow it as long as you suffer from waswas (obsessive whispering) and are following your brother's fatwa (definition). Your past prayers are valid, and you are not required to repeat them. However, we advise you in the future to take new water to wipe your head when performing ablution.