It would depend. I wouldn’t take in my ex-husband’s kids - but they are no relation to me and I have no contact with him (we never had kids) - there would be a ton of more appropriate people to raise his kids if they were orphaned.
But if my own kids have half siblings. Possibly. My son is adopted and there has always been the possibility - slimmer each year - that we will get a call that his birthmother has had another child. If she would want to place that child for adoption, the first choice would be our home. We have all the children we want, but if we got the call I think we’d make room for one more.
A toddler that was my kids half sibling - probably - a toddler doesn’t necessarily come with a lot of long term baggage. A teenager that was my kid’s half sibling from an ex who went out of their way to trash me - I think there would be better places - both for my sanity and for theirs.
I’d take her in and by the age of six she could start being a servant. She would nominally be educated along with my own kids, but made to understand her “place”. Then when my kids had children, she could step in and be the unpaid nanny. Eventually she can be my companion in my dotage (read - she can wipe my bum and feed me gruel). She should be grateful for the roof over her head!
I have to be honest, its easy IMO to say that I would take them in (and I am not at all saying that those of you who said they would are being un truthful) but I think I would be too bloody emotional and angry.
I can see a few instances where a person would not do it. Why they wouldn’t might be down to pride, as you say. Pride’s a pretty strong emotion.
There are any number of men in this world who abandon their own children. If you can do that, why would you take on the orphaned child of your unfaithful wife and her lover? Would anyone expect you to? Doubt it.
What if a husband and wife had no kids and she went off and had a kid with another man, then the parents both died. Would the husband there ask for custody of that child? And even if he did, what authority in the world would ever grant him custody?
That misses the entire point that this particular child is the sibling of his other children. There is a relationship there. In your scenario, there’s not, so of course the resolution for the child would be different.