Surely this would make the system worse!!
So you could have three groups of children in the same class coming and going at different times. That's surely having three different year groups in the same class. each semester group could have what 3 different levels of ability, so what the teacher is having to set 9 different sets of work?! Yes there's already composite classes but they're together at least the whole 9 months! The friendships develop, is that's going to affect the kids, they just make friends with some one and off they go to the next class!
Surely having students ready to take exams at different times in the year would increase the exam stress, again how do you prepare three different groups at three different stages within the same year surely that would make secondry school time tabling a nightmare! And they'd see their friends needing to revise and stress out over exams when theirs could be two semesters away!
How on Earth would this solve the holiday problem? what happens for incidence, if you have one child at year 5 one at year 7 and one in year 12, at three different schools and the primary school has weeks 3&4 and then week 10&11 the secondary school have weeks 5&6 and 12&13 and the sixth form have weeks 7&8 and 14&15 as holiday when the hell would they go on holiday?! The only option would be to take them out during their schools term time something thats actively being discouraged! What about child care it's hard enough as it is but partially at the primary/secondry turning point or if you have infant/junior schools or first/middle schools it could mean taking 4 weeks off work twice a semester!
I'm not sure you've thought this through and I've probably cross pisted with loads!