No, Shadow Blade deaths don't trigger grim Harvest
The question at hand is "does the spell cause death" when you're trying to trigger Grim Harvest. Short answer: No, the spell creates a weapon. The weapon causes death as a separate activity.
Jeremy Crawford clarifies that "death" is very specific to the source of final damage. Although this is aimed at "Death Savings Throws versus Knocking down to zero HP," it still gives us a nugget to consider:
Q: For Finger of Death: if the spell brings the target to 0hp and the target subsequently dies due to failed death saves, dies that count as FoD killing the target? Will it rise as a zombie?
A: A spell kills you if its damage or other effects slay you. If it reduces you to 0 hit points but leaves you alive, it didn't kill you.
Emphasis mine, but pay attention to that part. Does Shadow Blade's damage or effects kill the target? Well, the spell doesn't do any damage, it conjures a weapon. So the effect of the spell is to create a nifty weapon. So cast the spell and hold the weapon. Did anyone die? Nope.
Now make an attack with that weapon, via whatever abilities you have at your disposal. The attack roll and subsequent melee attack damage kills the enemy. Although the tool was used, it's not the direct effect of the spell that does the killing.
Consider this: Turn invisible and attack an enemy with 1 HP. Or 0 HP and two failed Savings throws, if it makes it more compelling. You only have one d20 in your bag, but you roll it and get a natural 1. But you have advantage from being invisible, so you roll it again and get a natural 20, hitting and killing the target. You only got to do that because of being invisible. Did being invisible cause the death? Nope. By the same token, creating a Shadow Blade is the effect of the spell. It's the attack with that weapon that causes death, which does not trigger Grim Harvest.