Regarding the deleted scenes...
There is normally a good reason why they are deleted, regardless of the production (*) - and the scenes from "The Warriors" are no exception. And in their case, redundance is the key.
The "group walk" scene contains an impressive, atmospheric shot of the gang walking through an ambiently lit narrow passage, but is overall pointless and redundant, with repetitive and mundane dialogue. If, on the other hand, it was cut to include *only* the passage take, it would become too short. (And it might provoke some to mischievously point out "Look, it's an obligatory _auteur_ shot!")
Cyrus's park speech is badly paced, needlessly extends his scene, contains mostly information that had already been made clear in previous takes, and does not add anything worth recalling. It is rather unconvincingly delivered, too - even considering the previously presented quality of Cyrur's take on thespianism.
"Cops and Furies" - once again, pointless repetition of the obvious. The viewers have learned by now that Masai receives every tidbit of information regarding the Warriors' movement quickly and efficiently, and they know everything that Masai is told. Redundant.
"Turnbull ACs" - this is one scene that, seemingly, could actually make it, as it is the very first instance of the announcer informing the gangs of the "progress" of the chase. However, its inclusion would have made at least one of her later appearances pointless - we would have already known that she relays the information via the waves, and repeating it ad infinitum would have become tiresome - and since all of her following takes have better pacing and tension, Walter Hill apparently decided to excise the weakest of these scenes... even if it was the first one. At least that's what I would have done.
"Where's Ajax?" - redundance, once again. We know everything told in this scene, so we might as well assume that Swan learns it from the rest of the gang, without having to witness the act. Additionally, the scene contains some rather stiff silent acting - hand me the scissors!
"We've got one more..." - redundant, redundant, redundant! We know all that! We know the Warriors do! We saw the Rogues' car trailing our protagonists! Cut.
And the infamous "daytime opening"... the bottom of the very worst. A nauseatingly dreadful idea that would not only destroy the braces of darkness framing the film, but would also spit in the face of the symbolic ending and set it on fire. As if that was not awful enough, the scene even comes with its very own insanely bad setup, infantile character introduction and choice of music which is, to put it gently, inane. The very possibility of including such a thoroughly wretched scene in the film is so frighteningly imbecilic that I am almost convinced that it was not Walter Hill's idea and that it was possibly shot without his knowledge or approval.
(*) - excluding instances of censorship and the sad cases of "marketing editing", obviously.
