There's no logical reason why someone would side with Songbird at that point. Selfishly, it gets you absolutely nothing personally. You now know siding with her will not only not cure you, but make you a powerful, vengeful enemy (NUSA)...for what? For someone who has lied to you and manipulated you every step of the way? Morally...she has deliberately and pre-meditatedly murdered hundreds of innocent people and is someone who is losing her mind bit by bit and if she loses it completely could end the entire human race by accident by causing a breach in the Blackwall? This is something that becomes progressively more likely the closer she gets to death.
Ok, I guess I need to add some infos here
For what reason do you think Songbird pierced the black wall?
Not for her own sake that's for sure... Songbird did it because Myers order her to do it. In the same way as Myers ordered her to "betray" Reed years before (something that has haunted her ever since).
So why does she die?
Because she pierced the black wall, following Myers order. So if someone risked the life of everyone on the planet by piercing the black wall, it's Myers and the NUSA, not Songbird.
Why she planned the whole shity crash?
Because for Myers and by consequences the NUSA, Songbird is a atomic bomb for them... She the living proof that by piercing the black wall, Myers and NUSA broke the international laws. Which can lead to another corpo war, but it would be the world vs NUSA. A war they sure will loose.
Myers keep a leach and a close look on her. Songbird don't have any other solution to get out, NUSA and Myers would never let her go (alive).
Either she keep doing what Myers order and die, either she try to escape, even with a shity plan...
Why did Songbird lie to V?
The same reason as V lied to Hanako. Hanako reveal you everything she know about Mikoshi because V agreed to help her... so except if you followed the Devil ending, V "lied" to Hanako too and at the end, she die because V didn't help her.
In my point of view, there are no reason to side with Myers/NUSA either
Yes, Songbird lied to V, Yes she killed people... But what do you think Myers will do if she get her hands on Songbird? She will simply stop what she did, she will become a "nice" president and will be respectful of the laws. No, she won't.... She specially asked to Reed to keep her head intact for a good reason... Continue her evil schemes and keep using the black wall as a weapon.
So again, nothing is full white or black here. But if you ask me and for what I gathered in the expansion, Songbird is the lesser evil compared to Myers/NUSA and by far. Again, depend of the angle you see it, but Myers will continue to pierce the black wall risking everyone's lives with impunity and without accountability to anyone. Something which won't happen if you send Songbird on the moon, so you possibly save more lifes than if you side with Myers. To quote the Slider (VDB dude) :
"If they break the black wall, we're all doomed!!!"
Where you get that? Blackhand siding with Militech on 2023 didn't end badly for him AFAIK.
Not that familiar with Blackhand "history", so correct me if I'm wrong, but we don't know what he really become.
He's alive? He work for Militech since? He fled? He's dead?
As far as I know, his fate is "unknown" until now
And even if he's alive, like Rogue he would be the exception... Everybody else involved in the assault and at the Atlantis were "exterminated".