It may not be as blatant as that. The verse starts out, literally, "And through this . . ." (καὶ διὰ τοῦτο . . .).
Depending on what you take "this" (τοῦτο) as referring to, the next part, "God sends . . ." (literally, "sends himself the God . . ." or "God himself sends . . .") could be understood or interpreted passively, even though the verb "sends" is active (and emphatic also). True, it (the NWT rendering) is a departure from the literal active meaning of "sends." But I think they are interpreting "this" to be referring to what Satan does in verses 9 & 10, and in effect, they are making God the 'passive allower' (if you will).
In effect, God has put Satan in a position to where Satan sees the need to pursue "this" scheme, and, having put Satan in the position where he needs to do "this", God simply 'allows' or 'lets" Satan proceed to do what was prophetically expected of him.
I see your point that the verb "sends" is active. And I do appreciate your pointing this out. It is interesting that the NWT engages in a bit of interpretation here. But I don't see it being all that significant in this instance.
Maybe if I look at your argument a little longer my view will change some.