Why can't I save new contacts to my phone or SIM? I just bought the quite new Moto G4+, which comes with Android Marshmallow (6.1) preloaded, without any layer added by the constructor. 

Upon the first use of the phone, I linked my Google account. When I want to add a new contact (via the Contacts app), there is no choice for the place where the new contact will be saved. Normally there are (at least) three choices: save (sync) to the Google account, or to the phone (internal memory), or to the SIM. Whatever I do, the contact will be saved in the Google account only. When I disable the synchronization with the Google account, the contacts are saved to the phone, but I have no option to sync a particular contact. How can I do to enable these three choices upon the creation of a new contact?

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that issue, there's an app named MyLocalAccount (from the makers of MyPhoneExplorer) providing such, but it seems no longer maintained (last update in 2013) – but after my comment with that, you were able to confirm it still does its job.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