In our school it is usually the TAs who are the most experienced and trained in specific interventions, and who are most often involved with specific outside agencies for certain pupils. Therefore, for us and our school set up - it makes more sense for the child requiring additional support to work with a TA.
We tend to do interventions in assembly time, and where not possible - short 15 minute sessions in an afternoon - so may miss a little time from class but never literacy, numeracy or phonics. They never miss PE or celebration assembly either, EAL or SpLD never miss singing or story either.
We do interventions when a child's IEP requires it, and with parental agreement.
We do a before school interention - also prepared and delivered by a TA - for children requiring support, usually Literacy, where prents may also attend to if they wish.
In a morning our TAs support the whole class learning in literacy and numeracy - and may be working with any of the groups, depending on the lesson plan for that day.
However, all this really depends on how your specific school works and how well trained/qualified/experiences the TAs are.
FWIW all our inspections and observations, from OFSTED, LEA adnd other agencies hoave rated our intervention policy as Outstanding and the pupil's progress (for those who attend them) as Outstanding - so something must be working right.
Our intervention timetable is very detailed and well organised - and, the key, is that our TAs are so well trained and work so closely with teachers but also, and rather importantly, the outside agencies.