Because Sites that use "if you play this Site" and Lost Souls that use "if put in territory" need to be triggered by the action that activates the ability, so all actions that activate abilities can trigger those abilities, if applicable.
Or, since Leviathan uses "blocks", not "enters battle", and we updated "block" such that it is a specific point after the first character enters battle (so a banding chain triggers Throne once), you could argue that Leviathan blocks after the ability activates.