I would tend to trust the LSE on the matter and their research suggested not
I think you are talking about their study looking at primary schools. Their research into secondary schools, where academies have been around longer, showed a different picture.
A 2013 report found that schools that converted to academies between 2002 and 2007 improved overall GCSE results by raising the performance of the highest performing students. However, they concluded that there was little evidence of improvement in the bottom 20%.
Another study in 2011 found a positive impact on pupil performance. They also found improvements in the performance of neighbouring schools, i.e. non-academies that have an academy nearby perform better than non-academies that don't.
There has not been any research that shows academisation is beneficial to primary schools
Agreed. However, the papers I mention above suggest that it takes time for the benefits to become apparent. If they are right, primary school academies have not yet been around for long enough.
Whilst I accept that some trusts may be charitable in aim, it is disingenuous to suggest that they are charitable
It is not disingenuous at all. It is entirely factual. Academy trusts are required by law to be charities. They are not profit-making businesses. Any surplus must be retained within the trust and used for the purposes of the trust. That may not be the case in your country but it is the case in the UK.
Just to be clear, academisation is not a silver bullet. There are good academies and bad academies. Even if one accepts that there is evidence that academies improve performance, one must also accept that some academies have failed to do so. And I would agree that the evidence is not yet there for primary academies, other than the (not entirely unreasonable) view that, since it appears to work for secondary schools, it ought to work for primary schools as well.