You are a very smart individual.
I am not practicing poly (yet), have two children, and a wife, and understand what you are saying about struggling with how much time could be taken away from those obligations by the introduction of possible other love interests. I love to play basketball, but if i play basketball five nights a week it's going to have an effect on my family. It seems reasonable for my family to allow for me to play one or two nights a week, because after all I am not ONLY here to take care of their needs. I have needs of my own that must be taken care of in order to be strong enough to help them in their down times.
Every relationship already struggles with those issues of time and care whether in the form of work, other relatives, school, play activities etc. If a person can successfully manage those, then they are capable of time management with other types of relationships.
It is admirable that you do not force your monogamy on your wife. However, you are an important part of that relationship and you have the right to stand up for yourself when you are being taken advantage of. Going off to Europe or other trips and leaving you as a single parent (if it's fairly regular and constant) is shirking her duties as a mother and a spouse. you are allowed to address that without saying we have to blow the whole thing up. While you say divorce is not a live option (and I feel strongly in my life that it's not either), I think you have to recognize that your relationship with your wife will have to change somehow. It is utterly unfair for you to "have no hope" while she is off with multiple lovers. It's not a matter of her problem being that she is poly, her problem is that she is being neglectful and uncaring which would be unacceptable in any type of relationship. If you really need a scapegoat and you want to make "poly" that, you may never get to the core of the problem so that you can have hope.
I wish you the best man. I wouldn't want to be in your situation for anything. But if your telling is accurate, you need to stand up and set some limits so you are not being trampled on. That's not right no matter what the relationship configuration is. If she doesn't understand that there is no such thing as total freedom when obligations have already been made, then she is not acting like an adult.
Is it so outrageous to her for you to ask her to slow down a little bit, or to put a limit on how many days a month/year she can leave you alone with the kids? You can't keep yourself in a situation where you allow her to ruin your life. Eventually that will not be helpful to your kids to have seen her do that to you. Take care of yourself so you can take care of them.