As regards Deputy Speakers in the Commons (but not the Lords), 'like the Speaker they lose party affiliation on taking up the role and must be politically impartial'.
I simply don't know if this impartiality extends beyond party politics for Deputy Speakers to, for example, the Referendum. The quotes I found earlier for the Speaker were more restrictive but this may have been the form of words rather than the substance.
My interest is less about the specific case of John Bercow and more about whether a precedent has been set for the future. Personally, I would rather have the Speaker impartial and seen by all to be impartial.
By the way, I do not understand what rules, if any, apply to ex-Speakers who go to the Lords. They can hardly be impartial and still contribute to a political discussion. Betty Boothroyd certainly made her preference for Remain loud and clear before the Referendum.
I agree that Hansard is useful and I have no reason to believe it inaccurate. It can, though, be tedious and boring.