My problem is that I come from a broken home and my biggest single goal in life was not to put my children through that - so I work at ways to make this marriage work
That's a oddly singular goal to have in your life and puts a lot of pressure on your partner. Its a commendable goal, no doubt, but not at the expense of the other important goals that should make up our lives. Such as having a loving relationship with your wife, being a respected and valuable member of society, being a well rounded person (which includes both being virtuous morally and hard working).
Now if you are a SAHP you are hard working, but so often problems arise where one partner is not contributing much financially. I think you have to be honest and ask yourself if this is at the route of the problem and if so, why your "primary goal" excludes working and providing financially for the family you are so keen not to break up?
I don't want to break up our marriage, my kids even ask me to make sure that doesn't happen! And as I said it would break my primary motivation in life. But I am very lonely and unhappy with my life, even though I also feel blessed with my children
Your wife's behaviour sounds awful (if a bit unclear - "talking" to other men is fine, if you mean she has actually committed adultery and you have proof of it then that's another matter). But at no point do you mention that you love this woman. You describe her entirely in terms of various roles that she plays in your life. And in fact you portray yourself as some sort of all caring, emotionally intelligent saint. It sounds as though she has lost respect for you and almost thinks that she can do anything and you will not leave. By surrendering your independence, you have lost control of the situation and by blaming her you are reasserting that loss of control.
Are your children of school age? Was it a joint decision for you to become a SAHP and if so, for how long? And why has it not occurred to you to separate or consider divorce if you are really that unhappy?
Before I became a father I was an adventurer, travelling the world with my work, not highly paid, but always exciting
That's an odd way to describe a job, unless you were a travel writer or actually a paid adventurer along the lines of Ranulph Fiennes, which would explain your somewhat esoteric writing style. If your primary goal was to keep your family together, a more certain way of going about it would be to establish a firm foundation first.
Basically, if your children are of school age, I think you should consider getting a job and establishing a career. If your wife really has lost respect for you, then that might help, but even so, it sounds like you should do it for your own self esteem. But do you really need other people to tell you this?