Sick or not, Alexei was spoiled. Any tutor would have had trouble with Alix as his student's mother. It wasn't the tutors who were at fault for his bad education, but Alix's for coddling him. While there was no reason for the Gibbes solution of letting him jump from table to table and kick lanterns, there were ways to let out that youthful enthusiasm and still get an education into him.
Alexei was spoiled, as was Anastasia, as according to some accounts was Olga. Their parents (BOTH of them), as the people ultimately responsible for their care, should take the blame for this. But - raising a royal child without an exaggerated sense of its own importance has to be a uniquely difficult task.
People often forget that Alexandra and Nicholas were not a minute-by-minute presence in their childrens' lives. They - and Alexei in particular - spent far more time with their nannies and later with their tutors. The whole world was lining up to ruin these kids: whether it be soldiers who saluted them, crowds who cheered them or passers-by who thought that a glimpse of them would cure their own families' ills - EVERYONE was showing them how important they were. And that includes parents who gave them suitcases emblazoned with their titles.
Sidney Gibbes was noted in previous employment for beating his pupils and being harshly critical of them. Yet (noted Frances Welch in her bio; it's not just my view as such) to him, the imperial children were wonders of charm and originality. Later, briefly, Alexei palled on him, but - as you note - he was himself always alarmingly liberal in what he let the kid get away with. In a nutshell: their tutors tended to be under the spell of their rank as well.
As I say, the parents bear ultimate responsibility for the way a child is trained, but Alexandra was not the person who was with her son every second indulging every whim. Other people played their own part in this. Including also - their FATHER. :-) And this isn't about who was to blame and whether she was a good or bad mother - rather the reverse: it's about saying: royal parents don't actually raise their own kids. Those cosy scenes in the mauve room are just one small part of the picture....