Could be. I think is more plausible though that we didn't get signings cause Woodward wasn't convinced that a) Mourinho is targeting the right players, b) Mourinho will be here next season, c) we couldn't get the players Mourinho wanted and failed to propose to Mourinho similar players with greater value. All of these things get fixed (at least in principle) by a DoF position.
Contrary to people's beliefs here, Woodward cares about United. Likely not in the sense of 'loving United like fans' but this is his biggest job he has ever had. If we do bad, then he is doing a bad job too, can eventually be sacked (he is not an owner like Abramovich who can do whatever he wants) and/or makes difficult for himself to get a higher position be it on the club or somewhere else. If United fails, so does Woodward. Assuming that he is a rational and intelligent person, he wants the best for United, not just to save face.