Why do most coaches save [time outs] till the end of the game?
To stop the clock for the possibility of getting the ball back and the opportunity to score, if necessary.
An example is from a 2013 wild card game with the Kansas City Chiefs at the Indianapolis Colts. The Chiefs turned the ball over in downs with around 1:55 left, and the Colts ran out the clock to clinch the win. Had the Chiefs had one timeout, they may have been able to get the ball back for the opportunity to score.
Are some coaches now starting to move their [time-outs] at "critical times" (e.g. time outs earlier in the game)?
This is where I challenge the statement, "These coaches still would have preferred to be able to save their timeouts for later."
The circumstance you present involves the Philadelphia Eagles offense and the New Orleans Saints defense. You also state that the Saints took a time-out to give their defense a chance to recuperate. This very well may be the...