I've been giving this a lot of thought, weighing up what I know of Walsh, his remit and responsibilities, how he comes across, our recruitment, continuity and the people brought in around him and while my knowledge of a lot of this is limited to just rumours and hearsay, I think my answer to the original question is yes, we should sack the sack of shit.
I should also say that a DoF should be strong enough to overrule a manager on a signing. They should be responsible for the long term, the vision and general style of the club. The manager coaches the available players to the agreed style. The manager is a short term, replaceable role in this structure and their opinion should be valued but not final.
If they're not strong enough to back down over the signing of a player that doesn't fit this, they're not right for the role.