Cannot add my own card when family sharing is setup

Index.xp wrote:

but that is a very heavy price to pay! for example, we share subscriptions to AppleTV, extra storage and lots of other stuff... why is apple making me choose? why isn't there a "Family and Private cards"?

It does not affect Subscriptions at all. It is only the content you purchase. The subscriptions can still be shared.

More info is here:

Share App Store, Apple TV, and Apple Books purchases with Family Sharing on iPhone - Apple Support

With Purchase Sharing on, you can also add money to your Apple Account using a Gift Card and any purchase will be deducted from that first before it will show up on the Organizers card. That may be an option for you as well.

You are free to use your own payment method as long as the Organizer is not using Purchase Sharing. Once they turn off Purchase Sharing, each family member can use their own payment methods. Purchase Sharing allows all members to share any app that was purchased, but is billed to the Organizer. Without Purchasing Sharing each family member only has access to the apps that they purchase themselves.

If Privacy is your concern, you should not be using Purchase Sharing. If one person downloaded a book, then everyone else would know what that book is and also read it. Shut it off to maintain privacy for the content you purchase.