Every NFL offensive coach has an up-tempo scheme in his back pocket. Some teams use it situationally. Some hardly ever. Kelley used it all of the time. The Colts once mastered it because Peyton Manning had the authority to call plays on the field and adjust on the fly. No NFL team has ever really done it better. The idea was to minimize situational substitution packages for the defense. Get plays called and run before opponents could get their defensive rotations in. Manning didn't do it obsessively but with nuance, picking and choosing the most opportune, logical times and catching opposing defensive coordinators off guard in the process. Hell, you can even take this concept back to Dan Fouts. The concept has been around since Don Coryell. Kelley didn't invent the strategy. What made Kelley unique was that he used it all of the time when the standard was to use it situationally. Kelley used it even when it didn't always make sense (to his critics). Kelley calculated that there was always more advantage for dictating defensive substitutions than there was for winning time of possession.
The biggest issue is clock management and self-discipline/situational awareness. You can "hurry" yourself into a quick "3 and outs" and put your weary defense right back onto the field without a proper breather (A similar clock management issue to the "run n shoot" offense that allowed for the biggest playoff comeback in NFL history; Bills verse Oilers). When your offense only consumes a minute of play clock for all three downs and then punts the ball away, you are giving the opponent extra possessions as well as conceding time of possession. Of course, you look brilliant with the strategy if your offense is humming, but then again, so does every "humming" offense. Chip Kelly has learned from all of this. He makes his latest iteration work because he started emphasizing a strong ground game which consumes more time and gains more clock control. Of course, the play calls will still be ultimately influenced by the type of opposing defensive package that is currently "trapped" on the field, unable to be substituted out. The whole idea behind gaining such an advantage in the first place.
The other big issue is good communication with the quarterback. The QB really has to be on the ball to receive the plays and get them relayed to the team on time. For that reason, I really wouldn't attempt the style without a veteran quarterback with plenty of autonomy to make his own calls and adjustments. It is an extremely difficult strategy to implement if every play is called from the sideline. It is why Kelley once used those cartoonish placards that assistants were holding up along the sidelines (University of Oregon). An attempt to make the play call communication more dependable and efficient.
In the end, you absolutely need an end game component where you can run your four-minute offense, maintain the lead by running the ball and keep the opposing offense off of the field. That was the big issue with Kelley's rapid-fire offense. With a lead, he kept punting the ball back to the other team without using up much clock. Keeping opponents in the game. When you condition your team to rapid fire plays, the transition to killing clock in the end game is harder to do. Regardless, Kelley has since evolved his style to compensate.
I actually think that Chip Kelley would be an effective offensive coordinator. I would be pleased with his addition.