You assume that the fruit always instantly leaves the moment of death, but it doesn't. Whitebeard stood there for quite some time without the fruit respawning instantly.
Smiley's fruit respawned the moment of death, but he also didn't just die - he completely evaporated on the spot.
It's possible that the transition period varies depending on how much damage the body sustained, means Burgess would have had enough time to bring Luffy's dead body to Blackbeard in time.