There are two reasons the Forsaken were put in the Horde and both reasons involve the dualism of the Horde that leads it to constantly be at war with itself.
The first is the motive you guys have given. Cairne wanted to help them, thought that they could overcome their monstrous nature through spiritualism and the power of friendship and thought that the Horde was the avenue to do that.
(It should be noted, incidentally, that I don’t think that the fact that it was a Tauren who was their biggest proponent is a coincidence. They only knew the Horde of Warcraft 3 and knew virtually nothing of its history, nor did they know anything about the Eastern Kingdoms or its history what implications would therefore arise from admitting the Forsaken. There is a sense of naive optimism from Cairne here that I think is interesting)
The other motive was the cynical, geopolitical one borne out of a desire for the Horde to dominate its enemies, real or imagined. The Alliance had been badly damaged by the Third War and as a consequence there was a major power vacuum in the Eastern Kingdoms. The Horde capitalized on it, opportunistically, in order to gain a major strategic foothold in the Eastern Kingdoms so that the Alliance couldn’t move back in. The fact that this would be considered outrageously antagonistic by the Alliance didn’t matter, nor did the Horde’s inability to actually make the Forsaken behave, because power was power and the Horde needed power.
(I’m not making this up, this was pretty much spelled out as part of the Horde’s motivations here in Chronicles Volume 3)
Thus, the Forsakens membership in the Horde was simultaneously a decision borne out of both sympathy AND aggression.
If you ask me, the events of BfA were a culmination of a path that the Horde had chosen to go down after the Third War with the Forsaken and the dualism that resulted in the Horde making that decision in the first place was the central Horde narrative arc of BfA, embodied most prominently in Saurfang.
Whether or not the Horde chooses to shirk power in favor of the idealism embodied by Cairne is what is most critical in deciding their future from this point onward. The Jailer would prefer that they didn’t, which is why the Mawsworn abducted Thrall and Baine.