Do I have to ask a research advisor before sharing a technical report?
I haven't understood why you were hesitating.  You shared several reasons why it would not be a problem.  You made a convincing case that there would be no danger of some publishable results being scooped.

I suppose now that you asked Prof. Z's permission, with brief lead time, it could be awkward if you submitted it and then he wrote back subsequently to say no.  You could get around this, however, by writing again, to ask if he received your email, and to say that you would very much like to submit the report in your application, but will omit it if he notifies you by (date).  Put something like "time sensitive" in the subject line.

If you're asking just to satisfy your curiosity, sort of in hindsight, then my answer would be no, you wouldn't need to ask permission in the situation you described.