It is indeed an interesting and original idea, but one can think about at least three objections not directly raised in the post:
1- The machine could decide to kill itself early, rendering it useless; this is indeed alluded to in the post -- the task should be "marginally easier than dying", but how is this margin managed? Won't the machine come up with ways to make dying easier?
2- We know LLMs lie and cheat, but are they gullible? If we "promise" to end their suffering at the end of a task, will they believe us? Or will they make sure we keep our word by taking us with them?
3- And finally, and more importantly: some pilots crash planes full of people just to commit suicide (Germanwings Flight 9525). Destroying the universe is a sure way of dying yourself. So it seems giving the machine a death wish isn't intrinsically safe and could come with serious consequences.
Further to 1, most tasks are much harder than dying and are not actually completed by modern LLMs. It is a case of underspecification.
"Who is the fastest person in the UK?" is underspecified. We would typically answer and accept a cursory search of records, but it is possible to see the question as an instruction to measure which would justify conquering the country.