Hello again, RC.
First off, *hug*.
Right now sucks. I've been in some suck situations that were similar as well--both in poly relationships and mono ones. Relationships can suck like that. They also can get better, or they can end. With kids involved, your relationship will never really end so much as change. Whatever your outcome, I hope everyone finds more happiness in the future.
As for the wife, even if she doesn't post her own story here, she might derive benefit from reading posts here. Some of what she sees in your posts may show her just how you feel and how you're trying to be a good man. She may see some things on dealing with poly life in general and NRE in particular that help her adapt. She doesn't need to post in order to benefit. As for her concern at you posting, you've shared no personal details that would identify you or her, so you're basically [mono guy Y] and [poly wife X] with [V partner T]. Even so, it's pretty clear that this group wants you all to be happier and healthier, so having a bunch of folks that want to help her help make things better isn't a bad thing.
That said, how do we deal with the present?
Everyone right now is likely in a heightened emotional state. Everything will be at extremes. You hurt more, you annoy her more, she annoys you more, T seems nicer, and T seems scarier. For those of us that have been there, these are true things--you're not alone in what you're feeling.
Probably the most important thing I can say is for you to focus on reducing your emotional level. You're an executive, so you know how to make tough calls. You work super hard, so hard work isn't too much for you. Right now you're faced with a bunch of conflicting data, and you want to make a call and start to move forward, but you'll be okay if you let this call simmer for a while. No jokes. You can work on the relationship or pull the plug in two weeks just as much as right now. But you want to be in a better place when you do. Your username is RobertCourage, dude. It takes a lot of courage to handle what you have in front of you, and I don't think you picked that name on accident. Sometimes courage means standing there and facing a fear without flinching while the rest of the troops muster up.
In the meantime, what does RC really love to do? Part of reducing your heightened emotions and stress is finding things that you really enjoy. The weekend is almost here. What can you do this weekend that would be fun? Forget about the emotions, forget about the talking and the worry, forget about everything but having a god time. Strung out RC isn't going to be as good a dad as chill and happy RC.
Will you have a chance to do something with the kids? They're at the age where a day at the water park or similar might be a neat way for you to bond with them, get some sun, and decompress. Whether the wife comes along or not is not nearly as important as RC getting some RC time. If that means you send the wife and kids off the see Harry Potter (part #zillion) while you wear boxers around the house and sing show tunes, go for it. (But please wrap up the boxers and show tunes before the kids get home--no need to scar them forever.
You're super focused on the family and the wife and the kids. What if you took care of just yourself and let everything else wait until you're in better shape? It's laudible that you're worried about everything, but sometimes worrying about everything undermines our operational effectiveness.