Is wiping the face three times, along with dripping water from it, considered a legitimate washing?

Wiping the face in Wudu (ablution) is not sufficient; rather, washing it is obligatory and is one of the fard (compulsory) acts of Wudu, based on the verse: "O you who have believed, when you rise to [perform] prayer, wash your faces..." (Quran 5:6). It is highly probable that what you described as water dripping constitutes washing, not merely wiping, and it is sufficient for Wudu, because the meaning of washing is simply the flowing of water over something.