Don't know if proper digs on controlling shareholders of Con Edison have been done.

Primary institutional holders:

Blackrock

Vanguard

State Street

BUT... the top institutional and top mutual fund holders don't seem to really account for a huge percentage of the stock.

Are there still huge private stockholders that wield enormous power in the company?

As the name implies, Consolidated Edison started with Thomas Edison. His major financiers in the beginning:

Vanderbilt

JP Morgan

https://finance.yahoo.com/quote/ED/holders

http://www.company-histories.com/Consolidated-Edison-Inc-Company-History.html