I'll dig up quotes when I get home
It's shown several times that the Black Sky is more of Alexandra's prophecy and obsession than any other Finger of the Hand. She made the choice to expend the resources.
Alexandra leaves a stray comment when talking to Elektra about how she had a daughter once. This daughter died (iirc, before Alexandra found the secret to long life). With continuous references to Elektra as her "child", there's an underlying implication that she feels more of a connection specifically to Elektra, and that Elektra is taking the place of her daughter in her mind/world.
When Madame Gao liaises between Alexandra and the Fingers, and brings up Alexandra's .... weakness, an interesting exchange happens. Alexandra specifically brings up that Elektra has something special, a strength that neither Madame Gao nor herself can match. This is cast as something unique to Elektra Natchios.
Elektra appears to have a bloodlust or "darkness" unique to her nature. It's not evident that any other Black Sky has it, or if this was purely a manifestation of the Black Sky.
It seems that (some of) these reasons act to motivate Alexandra, and she sees something in Elektra that we see no evidence of in any other potential Black Sky. Of course, it's kinda kept vague how Black Sky "works", or why the other Black Sky was being shipped to New York in the first place. The maps that Nobu uses and the radicals 黒空 around the block they wanted seem to indicate the Black Sky had some purpose there, but specific on how they intended to to "use" the Black Sky are unclear (were they going to kill him, were they going to try and use him to get the substance, etc.).
Alexandra seems to have thought that the Black Sky would just be generally useful in her hands, e.g. when it became clear that there was a wall and they might need the Iron Fist, the Black Sky could be tasked to get the Iron Fist.