This is US based but I think it is interesting as we can see parallels in the ways charities work across the pond.
The report that is discussed was commissioned by trans rights charity and details the public perception of trans activism and what is the best strategy to move forward. The point which is discussed the most is participation in sports, or rather the finding that there is absolutely no public support for it, even among people who they expect to be able to persuade with their arguments. This is followed by recommendations on how to approach the topic with policymakers which boils down to 'don't make it very public, target policymakers individually and pray the public won't notice'.
I would not be surprised if Stonewall's stance on 'no debate' originated from similar findings and realisation very early on that any debate will actually harm their cause rather than help.