Hi willyisaac,
If you'd like your family member to send iMessages, these steps might help.
You mentioned a "family member login" for the Apple ID. Is that a separate Apple ID or a single Apple ID that your whole family uses? If so, the following guidance will help.
Avoid sharing an Apple ID
Apple recommends that you do not share your Apple ID. Instead set up Family Sharing so that each person uses their own Apple ID. This way, your family can share purchases, subscriptions, and more without sharing personal information like email, text messages, or passwords.
If you share your Apple ID with a family member, certain services will treat you as the same person.
The information above is from Make sure that each family member has a unique Apple ID. There are also steps outlining how to create a separate Apple ID in the previous link.
If they do have a unique Apple ID, and are still having issues using iMessage, then let's continue. We see you've taken a look at If you can’t send or receive messages on your Mac. Did you have a chance to follow the steps below?
* Restart your Mac. Choose Apple menu > Restart, then click Restart. * Make sure that your software is up to date. Learn how to update the software on your Mac.* Make sure that your Mac is connected to the Internet. To check your Internet connection, try loading a page in Safari or another web browser.* Check that the date and time is set correctly on your Mac.* Make sure that you enter the correct phone number or email address for the contact.
If so, and the issue remains, let us know what happened when your family member signed into iMessages on the Mac. Do they get an error message when they sign in? If so, what's the precise message? For easy reference, here are the steps Set up Messages on Mac.
We hope that helps with your question and take care!