if the cash rich person would reasonably need to contribute to the other person, would you say they should also accrue some kind of interest in the assets?
This is where it does start to get complicated, because there is (a) the legal perspective, of entitlement and (b) the relationship you and DH have and your attitude to money (for which I hasten to add there is no good/bad, right/wrong judgement to be made, it is what you both need to feel comfortable and committed to, such that it doesn't become a future sticking point that sours the relationship you both have.
a) legally one would argue that if your DH already owned the house outright no mortgage, then it is his and he would be entitled to ring fence it. Particularly if heaven forbid you were to separate and divorce in future. I know you both decided not to go down that route, but it's a legal consideration. Also your DH would be within his rights to ring fence that asset and stipulate in his will that his DC would get the proceeds in the event of his demise. He could state in his will that you would need to die first, ie his DC couldn't sell your house from under you so to speak.
b) alternatively your DH could elect to give you x% interest in the property. Even if you have been contributing in terms of bills and rent, and even though you are married, that doesn't alter the fact that the default position in law is likely to be that he would have a major control and you would, in the event of divorce, have to take him to court, get a lawyer involved to fight for a % on the basis of marriage. I have seen this happen and believe me it is ugly. It sounds like you and your DH have likely delayed settling this, because of these rather grisly discussions on scenarios. I don't blame you, it isn't nice, however I would urge you both to have an open, honest conversation,to come to a mutually agreeable arrangement in the event of
A) death of either of you
B) divorce
C) protection and control of the assets you both have, and any provision you/he may wish to make to offspring. It is all about ensuring you both have control over what happens in the above scenarios, to protect the matter from being taken over by family members. This can happen, even if you can't imagine it now. The safest thing is to create wills.
I would suggest you engage a lawyer who has the expertise to guide you according to your wishes. They should tell you the law, but also work with you both, as you are perfectly within your right to deviate from the law, as that is only a framework.
Sorry for the essay, hope it helps you.