I have a Nexus 5X with android 7.0. A lot of my saved phone numbers are only searchable in the phone application, and are not listed in the contacts application. Various discussions, here and elsewhere, suggest that the contacts I can see in the phone, but not in my contacts app, are saved to the SIM card. However, both a third party SIM card management app, and attempting to import contacts from my SIM card to my contacts (via contacts -> settings -> import/export) indicate that there are no saved contacts on my SIM card. It is not clear to me where these contacts are actually saved then. And, how can I sync them into my contacts?
As far as I know, my default when creating new contacts has always been to save them to my gmail account (in fact, I don't seem to have any other option when I select "create new contact"), but somehow only a subset of those saved contacts actually seem to show up in my contacts app.