The same thing happened: After clicking "Change Plan," I landed on a page that only offered the option with a 3-month free trial. After entering my card details, I got the same message as shown above: a page with a red box saying I've used the free trial before but that I can still get Spotify Premium Family. When I click on the "get Premium Family" link, I again land on a page that only has the 3-month free trial.
The card should work, because I used it with Spotify in the past (that's why I couldn't use it again for the Family Premium Plan in the first place).
One of the troubleshooting hints was to try again in a few days. I again tried the link you sent me two weeks ago, and whaddaya know: it worked this time around! Thank you very much for your persistence.
I know there are other people facing the same problem that I did, so I hope Spotify will make your link the default for people who have already used a 3-month fee trial and want to sign up for a Family Premium Plan.
P.S. There is no "Accept as solution" button in any your posts. There is also no such option under the "three dots" menu. I sent you a Like in your post that contained the "winning" link, but I would like other people to be alerted to the solution as well.