At least, not before Zich explicitly rejected their recruitment.
They would not allow the Locke family to turn a sixteen year old champion with an SSS class summoning profession into an enemy, or destroy him before he could leave the First Layer.
As for why Kane's Life Lamp had merely shattered instead of being extinguished, the reason was not particularly complicated.
When Kane died, the third ancient beast altar was still completely sealed.
That trial space isolated every mental connection from the outside world. Even the Life Lamp mark the Locke family had placed within Kane's soul had been affected.
At the moment Kane died, the Life Lamp did not receive complete information regarding his death.
The connection between Kane and the Life Lamp had been directly severed by the ancient beast altar's spatial laws.
As a result, the Life Lamp did not go out as it normally would.
That was precisely why the Locke family still believed Kane might be alive.
