If I buy a lifetime annuity, and I pass on but my family did not report my death to the insurer or submit the claim for death benefit, does it mean they can get lifetime income?

No, it is not as simple as it seems.

The moment you pass away, all your assets will be frozen. Accordingly, there is no way for your family to get the money from the annuity. This is based on the assumption that the money is paid to your bank account (which will be frozen) or via cheque (which will be in your name and also frozen).

Additionally, there is a specific clause in the policy contract to state the need to report such incident to the insurance company within a stipulated period of time. This is alongside with other general provisions that prevent such incidents (that you mentioned) from happening.

On the whole, it is not worth as the repercussion outweigh the potential benefits.

Here is everything about me and what I do best.

Well this would be fraud, so the insurer could sue to recover... a lifetime of income, lived in fear of someone chasing you for money maybe not the best life.