According to ThoughtCo, the symbol of the Spades suit has long been associated with war, killing and death. In Italy it was depicted as a sword, and in French the suit was known as Piques – the word for a pikestaff. The symbol that we are familiar with in modern packs was standard in Germanic countries and can be thought of as an 'inverted leaf', but it also resembles the type of spade that's used to dig graves. A newspaper article reproduced on the PsyWarrior website (see below) describes the Ace of Spades as having been "long an ominous symbol in the poker–playing nations of the world".
The PsyWarrior newspaper cutting dates from the time of the Vietnam War, and this seems to be where the Death Card sobriquet originated. The Viet Cong allegedly saw the Ace of Spades as a harbinger of death. The Bicycle brand (a ubiquitous American brand) uses Lady Liberty in its Ace of Spades symbol, which the enemy reportedly considered to be the "goddess of death."
The United States Playing Card Company, producer of the Bicycle brand, was reportedly asked to supply American troops in Vietnam with decks that consisted entirely of Aces of Spades. If scattered in hostile areas, they were said to cause the Viet Cong to flee.
PsyWarrior has an extremely lengthy, and copiously illustrated, article on this topic. I haven't read it all the way through, but among the snippets I gleaned from it are that the Viet Cong's superstitions regarding the card were inherited from their former colonial masters, the French – but that many Vietnam veterans had never heard of its sinister connotations.
In conclusion: although I have no reason to disbelieve the stories regarding the use of the Ace of Spades as a psychological weapon in Vietnam, I suspect that its significance has grown in urban myth in the forty or fifty years that have passed since.
© Haydn Thompson 2019