What does the "Summon Dragon" button accomplish during the waiting period for Kairos to recharge, other than bring the Kairos Dragon back to its perch?
"Summon Dragon" (previously referred to as "Call") is a purely cosmetic behavior of the Perch of Kairos. Once you have acquired Kairos, he will either sit on the Perch or lazily fly around the area. If he is on the perch you'll see that "Summon Dragon" is not selectable. If he is flying around, pressing the "Summon Dragon" button calls him to the Perch. Within about a second he will teleport there with a time-dragony-wormhole-looking animation.
I don't believe Kairos needs to be on the Perch to use his "Activate" ability, so I see no mechanics reason to use this other than because you'd prefer Kairos stop flying around. I have seen a few account of him getting stuck, but it's unconfirmed if this ability can "unstick" him - and I highly doubt that is its core purpose.