Insomniac has worked with Sony numerous times, if Sony brought up the idea of continuing the SS arc, I don't think Insomniac would immediately turn it away
They wouldn't maybe immediately turn it away, but, when you've got a whole new team and you're suddenly forcing them to creatively confirm to another's work, that can be a sour note.
As others have said, Insomniac has the freedom to start from the ground up, craft their own world that will function as a launching pad for a new generation of Marvel games. They want to tell their stories their own way, but tying it to a show from so long ago invites problems.
I LOVE Spectacular and it was the last Spidey show I watched and cared about, but you're missing the bigger picture.
If they tied it to SS, they're limited to the show's continuity and machinations. (Heck, they may feel uncomfortable stepping on Wiseman and co's toes for such a thing.) This is a 23yr old Spidey and they may have completely structured his life differently from what SS established. What if Harry never became the Goblin in this, or they want a more comic styled origin of Eddie Brock, what if instead of Tombstone being responsible for certain events, they put in the Kingpin earlier to establish how big and long of a threat he is thus making it such a HUGE deal that he's brought down in the trailer of the game.
By making a brand new team have to adapt to an old work, you put a ceiling on their creativity. I'm a voice actor, so here's an example of creatives being boxed in:
Let's just say in a million to one chance, I get a chance to voice the Joker in the next cartoon show, or to keep things Spider-man related, I land the role of Hobgoblin or Venom, but with the stipulation that for Hobby, I have to voice him like Mark Hamil did in the 90s show, the show wouldn't have anything to do with the old show, it's just that the director loved that version of Hobby. That immediately boxes me into doing a subpar imitation of a grand voice actor and doesn't let me craft my own vocal interpretation of Hobby.
It's like getting free reign to make a modern Spider-man movie, but then getting told, they have to match certain things to the Raimi or Amazing continuity?
We don't have a lot of details on the story, but again, they're pushing for a young, but experienced Spidey. They've mapped what his internal history is and they may have certain villains appearing earlier or later and having different relationships.
In SS Norman was responsible for Rhino, Sandman and Doc Ock. What if they want to go for more comic origins for them in this? What if the Lizard only appears now and for story purposes it's to test Pete's maturity; having a 23 year old PS4 Spidey has him wanting to take a more measured approach in solving the case, learning he's Connors, curing him and protecting his family from a potential sinister force. 15 year old PS4 Spidey may have mostly fought mobsters and lower level Kingpin goons successfully, but then gets walloped hard by the Enforcers and Daredevil and others help him turn the tide. You couldn't have characters like Daredevil or Luke Cage showing up in SS, so why bar them from being part of this origin?
In launching a huge new direction for Marvel Interactive entertainment, Marvel doesn't want to take any chances, they want absolute freedom for this team they're bringing in. They want them to put their own stamp on things and go for the gold.
Insomniac is getting full autonomy so they're going to make things the way they want and don't have to budget time and money in going back, researching the way that Spectacular's seasons went down, don't have to look at legality with adapting any potential lingering rights, and here's a big one:
Bringing fans and new players up to speed.
Spectacular is one of my favourite cartoons and my fav Spider-man show. Yet it's been ages since I last saw the show, so they'll have to do something to catch up the audience and inform new fans of what happened on the show, what's happened in the 8 years since season 2 ended, etc. That's asking a lot of Insomniac.