Unable to share a purchased app to family members through Family Sharing

Hello. I (the 'organizer') have recently set up our Family Sharing group and one of us has a purchased app (Logic Pro X) that we are looking to purchase-share with me to be able to download it without needing to purchase it again. However, on my Mac App Store (following this Apple guide), when I click my account on the bottom left and try to switch the "Purchased by" account to the person with Logic Pro X I can only change to my own account and I cannot see others from the Family Sharing group - that is my issue.

I believe the problem has something to do with the 'Purchase Sharing' setting on the Family Sharing. I have not added a payment method because we aren't looking to have a singular payment method for everyone in the family. I have seen a few webpages from the Apple Communities discussions and Apple guides and the only thing I found was that I would have to do something related to changing the Apple ID that is linked to me on the Family Sharing settings (?).

In any case, ideally I would like to be able to simply download Logic Pro X onto my laptop through a shared purchase from another family member, something I currently cannot because I cannot see their already-purchased apps on my App Store (as stated above). Thank you.