As I understand it, I can only add new users to my Apple TV+ subscription if I set it up as a Family Share account.
An Apple TV+ subscription can be shared in a Family Sharing group of users with each their own Apple Account.
About Family Sharing - Apple
Learn more about what you can do with Family Sharing - Apple Support
Set up Family Sharing - Apple Support
Add a family member to your shared subscriptions - Apple Support
Is that correct, and is the only way to do that is if I and my other family members each use an iPhone?
It is not needed that each user has an iPhone, or any Apple device for that matter. But the organizer (the group member that sets it up) must do this on his or her own Apple device (iPhone, iPad, iPod touch, or Mac), so at least one Apple device in the group.
One user in the group would take on the role of Organizer, and invites the other prospect group members, each by their own Apple Account. Have them each create an account, if they do not have one yet. How to create a new Apple Account - Apple Support. This inviting should be done on the organizer’s own Apple device. The people that receive this invitation then choose to accept or decline. Alternatively, if needed at that time, some may choose to accept directly on the organizer’s device, instead of their own device. That’s it, then the Family Sharing group is established, and then the shareable subscription content will be available to all group members. Then each of the members get access through the own account; without the subscription account credentials themselves being shared. The subscriber and the organizer do not have to be the same person. Purchase sharing is an option within Family Sharing (with one group member paying for all purchases, presumably the ‘breadwinner’ of the family); that option doesn’t need to be activated for sharing Apple TV+ or other compatible subscription content.
Get the Apple TV app or Apple TV+ app on your smart TV or streaming device - Apple Support
Watch on these smart TVs and third party streaming devices - Apple Support
All the ways to watch Apple TV+ and more - Apple Support