Yeah, that would be the case. Whyte knows the deadline but he had to chase some time so that he can get something more of that purse but sadly he didn't got any. And Fury didn't ate the bait that Whyte have set because he doesn't care about those mind games at all and just like you said, he was just eager to fight and finish Whyte as soon as possible so that he can find another opponent to fight this year.
I got your point but Whyte have no other choice either but to just sign the contract and accept the deal that he is just only entitled 20% of the cut. Besides, there's actually no need to fret on those things because this is his biggest paycheck in his whole career. Also it is already kind of obvious that Whyte is just trying to get into Fury's head but the funny thing is Fury doesn't care.
Since this is the biggest paycheck of his career he needs to shape up and put all his best effort, he has so many things to gain and nothing to lose here, while Fury can lose everything in this fight, I don't rule out an upset because Whyte has knock out power but it can only come if Fury allows it, Fury is fast strong and has a good ring generalship but inside the ring, there's a lot of unexpected scenarios.
So the only way that I could see Whyte winning is that somehow Fury got overconfident and he did not trained as hard as he could while at the same time Whyte gave the best performance of his career.