Why can't I save new contacts to my phone or SIM?
As you've already found out yourself, this is a well known issue. Looks like your device lacks a separate "local account provider".

To work around that issue, there's an app named MyLocalAccount (from the makers of MyPhoneExplorer) providing such, but it seems no longer maintained (last update in 2013) – but after my comment with that, you were able to confirm it still does its job.

Unfortunately, I'm unaware of any alternatives to that. On a second thought: Using a CardDAV provider might work, but most likely will require a valid counterpart to configure. If you want to try that approach, you can find some in my corresponding app listing.