The youth, 15, crashed into two other cars at high speed before being arrested by armed police on February 20.
He claimed he drove off in the vehicle to settle a £40 drug debt, after mistaking the youngsters' father for a delivery driver.
The parent had been left crying hysterically in the street after his vehicle was stolen in the Bournville area of Birmingham.
The father told a previous court hearing that he shouted to alert the youth to the fact his sons, aged two and four, were inside the vehicle minutes before the crash on the city's busy A38 Bristol Road.
In a statement from the father, read at Birmingham Crown Court by prosecutor Syed Ahmed on Wednesday, he said the incident had been "the most horrific experience of my whole entire life".
He added: "It just made me sick to my stomach."
The teenager, alongside a 16-year-old co-defendant, admitted aggravated vehicle taking, incorporating dangerous driving, at a previous hearing, but had denied kidnap.
At the same court on Thursday, the youth was made the subject of a two-year rehabilitation order, placed under intensive supervision and surveillance for 12 months, an extended activity requirement for 180 days and an electronically monitored curfew between 8pm and 6am for six months.
The defendant, who cannot be named for legal reasons, was also ordered not to contact the victims' family, and his mother was handed a parenting order for 12 months.
Judge Peter Carr warned the youth it was in his interest to follow the rehabilitation requirements.
He said: "I hope you'll take advantage of it. If you don't, you'll go to custody."
The older child was uninjured in the crash but his two-year-old brother suffered a cut to his mouth and a bruised shoulder caused by the straps of his car seat.
The 16-year-old co-accused is due to be sentenced on Friday.