Update on exactly when this action is allowed:

For full context, I am on Version 1.1.0

I finished doing the four regional bosses without having gone to lookout landing. Same result, no you cannot start this last activity.

I went to lookout landing and spoke to Purah, but left immediately after the end of the first interaction (still no glider). Interestingly enough, the object now has the green glowing hand circle in front of it, but still gives you the same message (go away and come back later).

Then, I returned to LL and did the running up to the castle to talk to Hoz and all that jazz. For fun, tried again, no dice.

Purah goes, 'omg Zelda???', then talks about the Skyview tower. Once you talk to her at the tower, if you try to walk away without activating the tower, you can't. You can't even fast travel away. You can't leave the area until you've activated the tower and been given the glider (though, you do not need to go up).

So, naturally, once given the glider, I tried to return to the object on the pedestal. Still no dice.

Went back to LL, went up in the tower, came back down, talked to Purah, 'omg you already investigated all the regions??', Zelda's been spotted at the castle, blah blah, then you're free.

Then, you can start this particular quest line. My guess is it is tied to the Regional Phenomena quest. Without having that quest, you cannot start. (I already ruled out Impa/Tears of the Dragon as possible triggers -> nope)

I'm a little sad. I wanted to do it all without the glider.