Providing a home for DC is for most people an economic commitment, to be fair. So on an economic level, the GF is providing the home in the OP’s case. The couple are not married and the dad is not a stay at home parent to other DC, which I think would equal an economic commitment (so unpaid labour benefitting the household).
Whether DC are welcome or not in the GF’s house is an emotional question, which seems to be tied up for most posters on this thread with the ability to call said house home. As long as the dad lives with the GF it is his home, I don’t think anyone is disputing that. And no-one is saying DC are unwelcome. However, as other posters have also noted, this is not a stable or secure home (they are not married and he has no legal right to the house). It is not his house so whatever home he creates there is dependent on his GF’s consent to the relationship. Whether you agree with the OP’s depiction of the home as temporary or not, surely we can agree that dad has no claim to permanence in this scenario.
The OP has, as I recall, conceded that this is the DC’s home, at least temporarily. However, I would question whether DC actually do have two homes in this scenario, temporary or otherwise, or one home and one place they stay every other weekend. For all practical and legal purposes (presuming the mother has residency), their home is with their mother, their home address for school, doctor, dentist etc will also be with their mother. With the economic costs that entails.
I would say it is values based, rather than value based. Values in so far as providing a home is about more than making DC feel welcome but about providing security and permanency to the best of one’s ability.