Voicemail

Hi Type24.

Go to the section at the end of the article linked previously under the heading "Get help with Visual Voicemail."

Set up Visual Voicemail on your iPhone - Apple Support

If you try all of those steps, including resetting the network settings and you still do not have a Save option, you may need to restore the device in order to get the ability to set up this feature anew. As the Phone app is not something that can be deleted and reinstalled on its own, setting up as a new device will be the only way to set up that app fresh if resetting the network settings does not resolve the issue.

Please keep in mind that this step will reset anything network related to factory settings including Wi-Fi and VPN/APN settings so make sure you have those noted somewhere other than the iPhone before proceeding. 

Cheers.

Hello Type24 and welcome to Apple Support Communities.

I see you have a concern about saving a custom voice mail greeting.

Set up Visual Voicemail on your iPhone - Apple Support

It's important to note that only the Visual Voicemail feature is actually native to iOS. This is the voicemail feature that allows you to see a list of contacts who have left voicemail and access those in any order you choose. Everything else about voicemail—including saving and playing the custom greeting—happens via your carrier, not on the iPhone itself. You'll want to reach out to your carrier for additional support on this issue.

Cheers.