G&S are well known for their "hands on", cost cutting way of running a football club, so them getting rid of a member of the staff that picks up a decent check every month, and who is intimately associated with, and hired by, the sacked Duckers can't surprise anybody.
Zola came to the club lured here by Duckers and Nani and the "Football Project".
In the first interviews all Zola went on about was how important and interesting it was to develop young players and turn them into PL material. During that first year the most common theme in Zola interviews was the close cooperation between himself, Dux and Nani. That he was managing the team but that the three of them were working very closely together for West Hams future. This was also emphasized by Duckers and Nani, as in this Q&A with Dux on the OS. It was evident that this way of running things appealed to Zola and that he believed it could be successful.
After thrashing the results so far this season G&S were recently forced to either sack or back their manager. They decided to back Zola. For owners to go public with sharp criticism of the under achievement of the team tells me way more than any official backing following such a criticism. The manager is doubtless in jeopardy in that situation.
From the other angle Zola has lost 2 of his three mates in the management. Zola and Clarke are left to administer what is handed to them by the owners. This was obviously always the case, Zola had hoped for less Kovacoid players and more of the kind that Savio represented - at least until we saw him play... but at least he was involved.
Looking at some comments during the dying days of the transfer windows it is clear that Zola was not or didn't feel involved in the process. I talked about that in THIS post and the Zola quote I used was the pretty obvious "The Club are looking for some reinforcement but we didn't speak about a budget. They are trying to strengthen the team and that is enough for me".
Not involved and two of the three corner stones of the "football project" are gone. I think it is reasonable to extrapolate.