Hart was considered one of the top goaltender prospects in the 2016 NHL Entry Draft. NHL Central Scouting named him the top available North American goaltender in their 2016 midterm rankings. He was named the CHL Goaltender of the Year for 2016 on May 28, 2016. He finished the year as the second-ranked North American goaltender in the NHL Central Scouting final rankings.
Hart was drafted 48th overall in the 2016 NHL Entry Draft by the Philadelphia Flyers, the first goaltender selected in 2016. He signed an entry-level contract with the Flyers on October 2, 2016.
Hart was named both the WHL Player of the Year and Goalie of the Year for the 2017–18 WHL season after he compiled a 1.60 GAA, a 0.947 save percentage and seven shutouts. In addition to being named WHL Goalie of the Year for the 2017–18 season, he was named CHL Goaltender of the Year for the second time, the first being in 2016. He was the first goaltender in CHL history to win the award twice.