He was Hokage. He can give away what he wants.
If you're going to nit pick about the wording of the will, Kagome makes it clear that Jiraiya didn't mean to leave us
state secrets.
But okay. Fine, let's say that Jiraiya
did use his capacity as the Hokage to give away priceless state secrets to a new clan of missing nin from an enemy country, who haven't been in Konoha for a year, let alone long enough to have been naturalized --an act that would be a betrayal of the Leaf and everything that Jiraiya stood up until his death. Let's say that, for the sake of argument, that happened.
It doesn't matter. Jiraiya's dead. Dead. Gone. Dust and ash in the wind. Jiraiya's dead and now
Asuma's Hokage. Now Asuma can do what he wants because, by your own logic, he's Hokage.
What's keeping Asuma from having us all murdered in our sleep by loyal Anbu agents or village jonin, seizing the
very important information that is classified by the state, owned by the state, and
vitally important to the Leaf's national security and then funneling Konoha's far superior resources into decrypting it himself?
It's not a lack of ability. He has the full military might of Konoha to command and we have --at best --three chuunin, a social spec-ed jonin, and one sealmaster
against all of Konoha's military might. Sealmasters? Konoha has far more than one. Jonin? Again, Konoha has far more than just one, and theirs are
actually spec-ed for combat, not dancing verbally
(no hate towards Mari, I love her character bunches and want to see her get better). Even the Nara would have to concede against the full weight of Konoha, no matter how you put it.
And it's
certainly not politics. We've been in Konoha less than a year. We're not fully naturalized. We don't even really understand their culture. We're
outsiders, and all it would take is a small amount of rhetoric to convince everyone that it's okay, we were spies for Mist this whole time and a danger to the Will of Fire.
Nothing. Nothing, save the fact that Asuma likes us and is willing to give us a chance to work towards the betterment of our new home. He's lending us a lot of rope so that we can prove ourselves to be loyal citizens of the Leaf --something that I would bet my college tuition is still contested by a
nontrivial segment of Konoha's population.
Let's not hang ourselves in a killbox with the rope we've been given, okay? Okay.