I had this issue when I bought my first iPhone, (12 mini at launch, coming from Android), except I’m the family organiser, and I also set up a new iPhone for my mum at the same time.
There is no way for family members to independently use their own cards to make their own purchases.
It was a nightmare, and frustrating for both of us. It was so unintuitive that I thought I must be doing something wrong, but no, after contacting support they confirmed that only the family organiser’s card can be used by all members.
An option to use the family card makes sense, but disabling the ability for family to control their own finances and use their own card at all is ridiculous.
Mum can obviously manage her own finances, and managing their own money is absolutely something children need to learn too.
Equally important is the option for independence and privacy of choosing their own purchases without everything being reviewed by the organiser. Which is also not currently possible.
It was honestly a terrible introduction to the Apple ecosystem for the both of us, and really gave lie to the idea that “it just works”. Thankfully this happened to be the worst Apple experience we’ve had in the years since, and we’ve otherwise been mostly happy (family sharing disabled, of course), but it really wasn’t a good start.
And all I wanted was to be the family organiser to make things simpler for the less technologically inclined members of the family.