Direct supervision of solo activities, that is student pilots operating solo flights is all about the pre-flight activities. The supervising person is responsible to ensure the weather is adequate, the aircraft is serviceable and fueled, and that the student is capable of the flight and competent and understands the limitations of the exercise (one circuit, several, what to practice etc). Once the instructor jumps out of the aircraft the student becomes "PIC" of the solo flight and then is responsible as such for the flight. There is not much at all the instructor can do, or is responsible for once the student embarks on the solo journey. Any questions from that point will be back to the initial responsibility of whether the student was capable and competent to conduct that flight as "PIC".
As far as monitoring of the in flight portion, you just have to be on hand. It's probably a good idea to monitor the student visually on first solo, but subsequent solos, area solos etc, what are you going to do? It would not be legal to use a hand held radio on tower frequency, and would just cause confusion. These days you could follow the aircraft on flight tracking programs, but that is not required. If an aircraft is overdue you would contact ATC and try to find if it's safe.
Whilst I don’t know any that would head back inside during a first solo with eyes away, you certainly wouldn’t do it for a 15/16 year old. They are a child and you have a duty of care whilst they are in your hands.
No such thing as 'child' pilots, aviation regards a child as not yet 13 years old. Duty of care in regard to solo flights is the same and does not change if the applicant is 16 or 65. The main difference is under 18 the persons guardian(s) may have to provide consent to the activity. Generally legal supervision of minors is required for those under 13 years old, however this does vary between states some allow lower ages. School children traveling together have different rules and so on. Above that and the minor can travel and move around without a legal guardian and no one else will be held responsible for their safety.
Also it is commonly allowed for a 15-18 year old to be legally responsible for a younger child that would normally be considered too young to travel.