I don't think the fans are that easily misled. If they had hired Idzik and let him pick a coach, there's at least a complete regime change. And that buys some patience.
It really makes no sense that they kept Rex (if the plan is to fire him mid-season/end of season).
From a drafting point of view it makes no sense because you want your new coach to come in and have input on the future of the organization. For example Mangini got Gholston, Rex could do nothing with him.
From a fan point of view it makes no sense because firing both the coach and GM would have promoted rebuilding and would have gotten fans to temper their expectations of the team. No one expects this team to go far but keeping Rex and his annual promises of going far speaks differently from the rebuilding plan.
The only discernible reason I can gather is that Woody thinks that Rex can generate more PSLs than another coach, at the current time. NFL is a business, Rex is an entertainer. That being said, I feel like keeping Rex on a subpar team would have an adverse effect on selling PSLs and Woody's advisers would probably know that.
Not clear cut what the Jets plan is, but their actions speak volumes.