Apple is fighting me on not using either a password or Face ID

Are you using a feature that requires a Passcode:

• Stolen Device Protection. Turn it off.
• Apple Pay or card added to your Wallet. Remove all added cards/passes.
• iCloud for syncing content. Some content is encrypted and can only be synced with the Passcode to decrypt it. Turn off all apps for syncing at Settings > [name on top] > iCloud.
• Do you have any Hidden/Locked apps? Passcode is required to unhide/unlock them.
• Do you have any Hidden photos? Passcode is required to access the Hidden Photos album.
• Not sure what "pop-ups" you are referring to, if this is not the exclamation point in Settings, then a screenshot may be helpful. If this is for the warning in Settings to add a Passcode, then tap on it and press cancel to dismiss.

That warning would be appropriate when you have Messages in iCloud turned on at Settings > [name on top] > iCloud > Messages.

You are using a service that requires encryption using a Passcode and you don't have one to sync those encrypted messages.

Log some feedback with Apple: Product Feedback - Apple

Given your requirements, you’ll want to consider options for replacement gear, as well.

You are syncing your messages with iCloud, hence the password is requested or needed. Mac Jim pointed this out earlier. If you turn off all iCloud sync functions for all apps and data, this message may go away.

When I open text messaging, Apple keeps reminding me that I need to update my account settings (see below). When I go into the settings, decline face recognition, hit "don't use password", and cancel out, the pop-up keeps coming back. I'll follow up with AppleCare...thanks for your thoughts.

So you want to run around with your iPhone unlocked all the time and open to the world? Interesting.

lkrupp wrote:

So you want to run around with your iPhone unlocked all the time and open to the world? Interesting.

HomeView might be an alternative, depending on the (other) requirements.