I would say it is, especially considering this sentence in the introduction of the wikipedia article you link:
[…] The goal of a white elephant [gift exchange] is usually to entertain rather than to gain.[…]
I'm taking part for fun, certainly not to get anything special or valuable.
Until now the bigger part of the fun was choosing a gift for a complete unknown not supposed to know his Secret Santa,
the incident part of it was receiving a gift from a complete unknown not knowing who is giftee was.
Knowing who his giftee is puts pressure on any Secret Santa, as very well shown in the discussion above: now nearly everyone tries to find out what his giftee would like to get, and as a consequence even expresses own wishes.
To me it's the complete opposite of fun.