I kind of agree that, as a general rule, financial responsibility for a child should rest with the actual parents and not the step parents.
BUT there are certain caveats to that.
For example, if the parents can't or don't earn enough to actually support the child, then I think a resident parent's income should be taken into account before expecting the state to step in. It would be ridiculous for the taxpayer to pay welfare benefits to a mother on a low income, for example, if she happens to be living with a millionaire partner and enjoying all of the lifestyle benefits that this might provide.
I think it would also be fundamentally wrong for children in blended families to have very different standards of living from one another because of what their parents can respectively provide. In that set-up, I would say that there is a duty on both parents to provide equally for all children in the household, and if they don't want to do that, then they shouldn't live together.
Of course, non resident biological parents should be forced to pay their fair share, and that should always be the first port of call. But I don't think that entirely lets step parents off the hook.
Basically, if you don't want responsibility for someone else's child, don't move in with them.