Oh god, no. That idea is dead in the water. 6 pieces set, 800% damage (or something) for every set item equipped. It’s an LoN for set builds, but you can use up to a 2/4 piece of other sets.
How is that any balanced compared to LoD? You can still combine Aughild’s or Crimson’s if Blackthorne’s is a 6-piece only Set bonus. Damage bonus need to be diminished and it would never be truly balanced compared to LoD. You might dislike the idea of not having capable of speedfarming with LoD/LoN due to not being able to equip Cain’s, Sage’s or whatever but that’s the distinctive characteristic of the LoD or LoN Set anyway. I hope it’s clear.
It’s not that much more powerful than LoN, and may in fact be weaker. LoN can utilize affixes on every slot. Some of the most powerful affixes in the game are legendary - not set (Cough, CoE). You wouldn’t be able to mix and match affixes like LoN does. You would pick up a 2/4 piece set from something else (it would depend on the set). You would gain an additional Legendary gem, but you’d probably have to give up CoE.
But it’s actually reasonably balanced.
The class with the most benefit would be the Necromancers (of course), with their 400% damage 2 piece weapon sets. So you’d have pseudo-LoN necromancers running around with 400% damage on their weapon sets, but they’d have to give up a weapon/offhand affix.

It’s not that much more powerful than LoN, and may in fact be weaker
Perhaps. Still both LoN and your suggestion can utilize damage and skill procs as both increase overall damage. There must be a catch to the model you suggest. Perhaps it could only increase the self damage and never it gets reflected to the pets, devices or clones but that’s doubtful too.

The class with the most benefit would be the Necromancers (of course)
There are more than one 2-piece weapon Set bonuses; Norvald’s and Manajuma’s comes to mind besides Jesseth’s of Necromancers. I don’t think they’d need more buffs as top builds already utilize said Sets.
Also on some classes, I can imagine it being hard to foresee when you consider something so simple as Shadow Set granting 6000% damage increase passively by 2 pieces bonus.

(Cough, CoE).
That’s not a constant 200% damage increase but a cycled burst, you can not hope to balance the idea on 100% efficiency of an item without minding uptime or average player skill.
Another 800% more damage or whatever over 12 piece of bonus as the 13th Set piece only budge the overall damage for +8%. See here for the math: [(13 * {800%} * 100)/(12 * {800%})=108~], so CoE would be a legit piece of that Set idea you have regardless of what you foresaw.
CoE is not really an edge over the suggested Blackthorne idea as it can easily integrate it to the build. You can never really land a true balance by tweaking numbers for the model you propose.
If it’s going to be a Loan for set builds it should be the only set. Provide a huge buff for being the only set equipped with all 8 pieces.
If there’s ever 8 BT set pieces in game. And I don’t think there’s ever been an 8 piece set. It might be a helluva mess.
Aren’t there 8 items altogether?
If so, I wouldn’t mind seeing it as a
boots, belt, amulet, pants, chest. I believe there are only five.
Correct only five items in the set.
I have full primal set of blackthorne on my barb, gr100 is piece of cake!
Well, I was running frenzy but with full blackthorne primal set.
Check out my character “LovePrimals”.

No. With the they OPness of the way it was initially described, make that 15,000 bonus an 8pc.
15,000 would probably make sense as a 6 piece set. I’d take that number.