DS does Trinity Jazz syllabus for clarinet and saxophone - because his route into music was via jazz.
However both the county senior jazz band, and, perhaps more surprisingly given Pirate's comments, the county senior youth orchestra, are equally happy to accept his qualifications as showing he is of a high standard.
It seems to me that the technique you acquire is down to your teacher - a good teacher will teach correct technique whatever pieces it is applied to.
Where the two differ - noting that DS has chosen the jazz route for Trinity, not the classical one - is the range of skills he is required to / has the choice to show on top of the pieces. Studies based on scales / arpeggios rather than scales / arpeggios themselves. Musical knowledge and improvisation as options rather than aural tests. The style of the sight reading.
It seems to me that, as long as the teacher teaches sound technique, the choice of board is down to the preference / target of the child concerned. I also have a dancing daughter, who does ISTD Imperial Classical Ballet (the 'Trinity' option of the dance world) rather that RAD (the ABRSM option). The grades are equivalent, on average - harder at some points, slightly easier at others - but one depends critically on the knowledge and skills to be able to dance unknown dances set on the day by the examiner, the other on absolutely perfect execution of standard set exercises. They are not 'greater' or 'lesser' skills, just different, and will give the examinee a slightly different skill set in their future life.
My understanding of Trinity / ABRSM is similar - a jazz musician doing Trinity will have the skills needed for jazz playing - improvisation etc - to a greater degree than a musician doing ABRSM. However the latter will have greater experience of strictly 'classical' aspects of music such as aural tests. Which a child does should be dictated, to an extend, by their musical ambitions.