Given how the Meteor was presented as Hoshigakure's secret-possibly-ace-in-the-hole in the chargen, I think it's reasonable to guess that it's a special prize. And those aren't easy to get. Or at least shouldn't.
Given how Chiyome and Endo, who are quite learned in Chakra Studies, have never heard of the Tree, I don't think verifying its existence is going to be easy. Maybe I'm wrong and the spiders will straight confirm that it exists, but I expect we'll just get clues that we'll have to hunt and check.
The third Hoshikage won't accept anything else that 100% proof, which seems another factor that will increase the amout of work the project will take.
Since the former way to train Meteor nin was faulty, having to re-design it doesn't sound far-fetched, adding yet another step.
I admit that I never thought about delegating Meteor research to other councillor, but something like that doesn't seem to fit the narrative (we're Hoshigakure's top researcher, very involved in it, and what happened to Chiyome's father made it personal) or the game (investigating the meteor is a goal for the players), unless most questers get bored with the plot hook and ask to drop it.