If Seymour indeed did become Sin then yes, with his philosophy it would be end of all life on Spira.

When Sin got out of control he was always provoked, Mi'hen Operation, attack by main party etc.

When he attacked the ship or Kilika sure he was scarily powerful but he didn't obliterate those.

And I am pretty sure it was just those five high summoners, there is no mention of any other and no other indication of them.