The down is 100% the responsibility of the officials, not the coach. because that's what the referee said it is. The officials are the SOLE keeper of the correct down. Coaches have ZERO responsibility for the down. In legal terms, its called "detrimental reliance".
Thankfully, we're not in a courtroom, where far too often people are allowed to get themselves tied up in knots parsing words to mean whatever they want. We operate on interscholastic athletic fields, where hopefully players learn about accepting personal responsibilities, rather than deflecting them to others.
Coaches are SOLELY (usually) responsible for selecting which plays to call, which
REQUIRES understanding the correct circumstances they're playing under. Knowing the circumstances, which includes questioning them when there is doubt, is the Coaches responsibility, and
failing to do so rests with him.In absolutely
no way does that absolve game officials for failing to correctly satisfy their requirements of displaying correct information, but that is a separate failure,
subject to separate and distinct consequences and it
DOES NOT ABSOLVE the Coach for neglecting his responsibility of
KNOWING the correct circumstances for which he is responsible to select the proper play for.
We all pursue perfection, and few if any
EVER catch it. A major part of the game of football involves how we react, respond, learn from and avoid repeating mistakes. Mistakes
WE make, even those resulting from the actions of others are
NOT THE FAULT of those others,
OUR mistakes are caused by
OUR failure to react and respond to the circumstances we are presented with, and responsible to adjust for.
This interpretation, creating "do overs" does NOT improve or enhance the game, and hopefully will be corrected and reversed.