Pick one list per person, or one for the family
A list for each person is clearer: send each link to the people who buy for that person.
A single family list is faster: add gifts and put the name in the note, such as "For Sam - blue scarf". Fewer links means fewer messages to lose.
Set the budget with the prices
Add gifts at several prices so each person can choose what fits. A saved price also tells the buyer what to expect before they open the shop page.
Check the shop before you buy; the price is from the day the gift was added.
Add links early
Popular gifts sell out. Add the link and the details while the item is in stock, then check again in December. If a shop stops selling the item, keep the note so a buyer can find a similar one.
Use the reservation to keep surprises
A guest reserves a gift and the list shows it as taken. The name of the person who reserved it stays hidden from the list owner until the owner reveals it. That keeps the surprise for the person who is opening presents.
Send the link once
Put it in the family group chat or the message where people ask. Say what it is and that there is no pressure. Do not send it to people outside the group.