I am a governor in a primary school and I read this research with interest, because we have a team of TAs who I regard as awesome. However, I do think there are key issues about how they are recruited, trained, supported and managed.
I read the research linked to by OP as saying that children receiving most TA support made the least progress. The suggested cause was that these kids were being effectively taken off the class teachers watch and receiving much of their teaching directly from the TA. This creates a double whammy in that they are losing out on the whole class teaching (if being withdrawn) and are getting their teaching not from the person who is most highly trained and qualified at doing it. Interestingly we found a similar effect at a time when lots of our pupils with SEN were withdrawn to work directly with the SENCO. DS was one such child - it may have been a coincidence but when they stopped doing it, his performance sky rocketed.
The research also notes the value TAs bring (as some pps have identified) in helping kids to stay on task, prevent disruption, and generally making the classroom run smoothly. Essentially making sure the teacher can teach without interruption and that kids can access what they are teaching. You won't measure that as TA input per individual child of course.
We have very well qualified, probably ridiculously over qualified, TAs. The Deputy Head has put in a management, appraisal and development structure for them. They're all encouraged to specialise in areas such as EAL or particular SEN (eg my sons TA is also the school ASC 'specialist' TA). We try and hang onto them having recruited and developed them. If they run interventions, which they do, they are carefully planned, directed by SENCO, and there is evaluation of whether it worked or not.
What someone needs to try is comparing schools with similar intakes and either the well trained, supported and managed TAs, the 'bog standard' TA model, and no TAs at all - then test the effect on results.