I have done both the required quest The Far Edge of Fate and A Flare for the Dramatic. But Y'mhitra wont give me the next quest, A Book with Bite quest, for the Summoner? I have had no issue with the other job quests lines. Im lv80, the quest is lv60, and about the start the Shadowbringers Main Scenario Quests.
Double check that you were in the Summoner job, with the Summoner's Job Stone equipped.
If you had happened to been switched to the Summoner job, but didn't have the Job Stone equipped, then your job would have actually been Arcanist, which would then remove you from the Summoner job quest eligibility.
One common way to have your Job Stone "unequipped" is by changing jobs via the Weapon Swap method, basically either by hot-key/hot barring a weapon or by drag/drop a weapon from your Armory inventory.
Summoner and Scholar are linked jobs, as in you level up both at the same time, but sometimes when people are quickly switching between the two jobs, by using the "Weapon Swap" method, they do not catch that the Job Stone wont auto-equip, resulting in being "downgraded" to Arcanist. This is because the weapon for Summoner is also flagged for Arcanist, while the weapon for Scholar is only flagged for Scholar.
To avoid that, set a Gear Set to that Job (which has the Job Stone also equipped), and switch via that menu.