My interpretation of the ending to 9 is that it's the beginning of the Final Fantasy universe/multiverse.
In Memoria, you keep going back in time until you get to the Crystal that is the singularity before the FF big bang. When Kuja Ultima nukes the singularity to try and end everything, the party wakes up on the hill of despair which is an outside of time and space purgatory nothingness thing.
There they encounter Necron as the embodiment of death. If the party loses, the crystal is destroyed and the big bang doesn't happen, or even have ever had the capacity TO happen. If the party wins, Ultima hits the crystal and big bangs the FF universe into existence.
This is also a way to tie in all the nostalgia throwbacks cleverly to the story. Like the buster sword being on the wall of the weapon shop in Lindblum wasn't just a fan service callback. The FF9 world was shattered to become the rest of the universe/multiverse so the FF7 buster sword was the "memory" of that weapon from 9.
It's actually pretty ironic that Lenas turned the game off and quit after losing to Necron once as that was symbolically the characters' dilemma also. Would be even more fitting if you never played any FF ever again lol.
As far as the story importance of summons, they are either very important to the world building or absolutely essential to the stories in 4, 6, 9, 10, and 15. The rest they're mostly a common theme. In some you just find Leviathan in the basement of a weapon shop or something and no mention is ever made. In 6 the entire esper race and their relationship to the warring triad is a huge part of the plot and every esper has a place in the story. Many are still inconsequential but at least were part of a research experiment or in Ramuh's party that gave up their lives for you. Then Terra is half esper and it's her entire character arc with her parents, her dad being temporarily playable and then later summonable.
Totally down for 17 being the tie in to 10 and 7. They're already a part of the same solar system/universe and with how beloved both games are to fans, would print money.
Us FF lore nerds showing our colors in here. It's still a JRPG series so it doesn't make total sense but it's not as bad as you're all saying.