The wicket-keeper batsman has averaged 20.25 in red-ball cricket for England this year and scored only one half-century in the Ashes.
It has resulted in the Yorkshire player missing out on a place in Joe Root’s 15-man squad, which contains uncapped foursome Dominic Sibley, Zak Crawley, Saqib Mahmood and Matt Parkinson.
Jos Buttler will don the gloves while Surrey’s Ollie Pope will return to the middle-order after two tests against India in August 2018.
England play five T20s in New Zealand, which Bairstow will be involved in, but with the two Tests not counting towards Test Championship points, the selectors have used these games as a chance to look at others.
“Jonny is a very talented cricketer and we believe he has the potential to be a top, top player in Test cricket for England,” Smith said.
“He has played a lot, in all forms with a very hectic schedule and it’s an opportunity for him to reset and work on one or two things.
“If I was asked to give a prediction, my prediction is he will come back stronger and have a very good England career in Test cricket in the future.”
Bairstow’s omission will not be the only change to the team which defeated Australia at the Oval in the fifth Ashes Test to draw the series 2-2.
Warwickshire opener Sibley is expected to open alongside Rory Burns, while Joe Denly should bat at three after captain Root asked to return to his favoured position of four.
Smith explained: “We obviously work very closely with the captain, he wants to bat four, we want to support that so we want to develop a top three.”
Given Eoin Morgan had a World Cup to prepare for and Root had the Ashes, the two England captains have not had many chances to witness the best players coming through on the county circuit this summer.
Despite this, Smith revealed the pair were a big part of the selection process.
“Joe and Eoin have been very involved, even though they have had a long summer,” Smith said. “We wanted to give them all the information we could obviously from our own eyes and from the scouts.
“Even though they are both out of the country currently, they have been peppered with phone calls. The specialist coaches were involved and the same applies to Graham Thorpe and Chris Silverwood.”
Sibley and Crawley are rewarded with Test call-ups after excellent County Championship seasons for Warwickshire and Kent respectively.
“It’s a thrill to have it confirmed and to be going away with England to New Zealand is really pleasing. We’ll see how it goes out there and hopefully I will get a Test match,” Sibley said.
Crawley admitted to expecting a Lions rather than a Test call-up, but added: “It was an amazing phone call to get. Once he said I was picked in the full squad I kind of zoned out for a few seconds and wasn’t totally sure what he was saying for a moment, but Ed was very positive and I can’t wait to get going.”
Mahmood and Parkinson have enjoyed more success in white-ball cricket for Lancashire, which has resulted in them being part of Morgan’s T20 squad, but they will remain in New Zealand after the five T20s are over and be part of the Test set-up with England thinking “long-term.”
Parkinson found out the news via his team-mate Mahmood shouting down to him from the balcony mid-match.
“I’m very happy and proud, and it’s an amazing way to finish the season,” he said.
Jason Roy is not part of either squad, but Smith insisted that was down to an “incredibly hectic summer” which also explains why Jofra Archer, Root, Ben Stokes, Buttler and Moeen Ali will not play in the five T20 internationals.
With the T20 World Cup just over a year away, Morgan will cast his eye over Somerset batsman Tom Banton and Worcestershire fast bowler Pat Brown against the Black Caps.
England will be without James Anderson, Mark Wood and Olly Stone for the Tests in New Zealand due to injury, but they may be part of a bowling camp which will act as preparation for the winter tour to South Africa.