Answer
If the leader crashes or is cut off, nobody has to prove it is dead: the lease simply runs out and another node can take over. The leader renews it well before expiry and must stop acting as leader as soon as it cannot, on its own clock, be sure the lease is still valid.
Why
Leases depend on clocks running at about the same rate, not on synchronized clocks: the holder gives up a little early to allow for drift. They do not protect against a holder that pauses and wakes up after expiry still believing it leads; that needs fencing tokens.