I'd put Andrew Garfield as Peter Parker and someone as Miles Morales
Keep the Amazing Spider-man continuity. Just push the timing back, so it happened before the Avengers despite 2 releasing after it.
That way you get an older Spider-man so you can stick J Jonah Jameson in there (same actor as the "original" trilogy). You also keep the side characters, because I basically thought they were all really well cast
But at the sametime, you can move on from the franchise a bit and give a fresh new face to Spiderman
I'd also be thrilled with Grant Gustin. Even if him as the Flash at the same time will be confusing (especially as there will be another Flash)