There’s emotional conflict built into this card’s position - it sits right next to the Ace of Hearts in the natural order, so you get this deep desire for love constantly clashing with mental rigidity.
The karma card for King of Spades is the 5 of Clubs, which is about restlessness and constant mental change. That tracks with what you said about this card being strategic and always calculating - the shadow side is this inability to settle mentally, always needing the next challenge. That tension between mastery and restlessness is kind of the whole story with this card.
One more thing. You mentioned Kings carry the number 13 in numerology and I think that’s underappreciated. There are exactly 13 cards per suit, the King as number 13 sits at the transformation point, and the Death card in tarot is 13 too. That parallel between mastery and endings is just baked into the structure.