Can NFC be used by iOS applications, e.g. SSH authentication via smartcard, where the private key never leaves the card?

Not at the moment, but one element suggesting it will be in iOS 13 is that the UK Home Office announced their app for validating European passports (which uses NFC but only on Android at the moment) will be able to work on Apple devices at the end of this year - basically when iOS 13 will be released.

I think it is possible in iOS 13, unless I misread the WWDC video description.