What's a better way to think about it?
Ok, a couple of thoughts (please don´t take any of this personally - you asked):
1. "ownership", "a right", "responsibility", "duty" etc are taken from "Amtssprache" ("beaurocratic language"), the language of power, pressure (and violence) - representing this very of approach, attitude and thinking; a particularly ugly language that doesn´t belong in our private lives. Our personal relationships aren´t about distribution of power.
2. Saying that your children "have a right" or "own you" is factually inaccurate. It is being a pompous way of saying that you feel they are in need of care and protection and that you feel inclined to give it to them (which is a good thing), out of your own volition. Children do not have such a right or ownership - they couldn´t even claim or enforce such. It is
you who offers them this or that, but that doesn´t equal a "right". It doesn´t change anything about you being the factually powerful person at whose mercy your children are.
You are the one who determines what responsibility to take. In my understanding, for there to be a "right" or "ownership" there must be a third party authority granting these (and helping to enforce them). In picturing your volitional decisions as handing out "rights" you are announcing yourself party and judge at the same time.
3. That your children "own" you is also inaccurate for the fact that while you feel they have "a right" to demand that you abstain from suicide, you certainly don´t feel they would have "a right" to demand you to commit suicide if they so wished.
4. Dealing in terms of "Amtssprache" is a dangerous thing. As Marshall Rosenberg (inventor of "Non-Violent Communication") puts it: Everyone involved will pay a high price of this. (Example: Typically, this generous handing out of "rights" to your children will eventually result in you claiming "rights" for you and "responsibilities" for them, once you get old.) It is establishing the idea of mutual pressure as the basis for our interactions.
Now for the alternative:
You start thinking of human relationships and interactions in terms of empathy, well-being and fulfilling needs (which include your own needs).
Caveat: If you feel you are not empathy gifted this approach obviously won´t work for you, and you might be left with the approach of mutual pressure.
Do you
want to commit suicide? or
Do you
want to abandon your wife and/or your children?
If no: How do questions of "rights" and "responsibilities" even come into the picture?
If yes: We may now start to empathically look at the needs of everybody involved.
And the first question that comes to me:
"To which extent will the needs of your children be met if you stay alive although being desperate or staying with them although it makes you unhappy? What are the alternatives? What - seeing that having a father who stays alive (stays with them) just out of a self-imposed or allegedly external pressure might not be a particularly good environment for children (who need honest care, positive emotions, parents who have a positive outlook on live etc. etc.) - other solutions could possibly account better for the fulfilment of the needs of the children?
and next:
"Isn´t knowing that your father has spent a life of unhappiness and lack of fulfilment due to his perceived responsibility towards you an incredibly heavy burdon to have on your shoulders when starting into life? Won´t that make you feel unwelcome forever? Won´t that put a pressure on you automatically (even though no return has yet been demanded)?"
I´m not pleading either way here (a lot depends on the individual cases and circumstances), and a father who commits suicide while you grow up certainly isn´t a pretty thing either. That´s not the point. The point is that I am pleading for a different way of approaching the question.
If sometimes: There is a good chance to find a solution that accounts for the fulfilment of both your needs (which probably involve more time for yourself) and your children´s needs.
On a final note: Once you notice the desire to commit suicide or abandon your wife and/or children, you may want - instead of escaping into ugly, incorrect, pompous "Amtssprache"-thinking which is likely to perpetuate your unhappiness - start to consider how it has come this far, and if there is any chance to restore a life in which you can happily coexist with your wife and children (instead of responding to self-imposed or allegedly external pressure).