It's not really a subset:
MIFARE Classic uses a proprietary and mandatory encryption/authentication algorithm and is therefore not ISO 14443-4 compliant. As a result, NFC-compliant readers don't need to support it, and in fact non-NXP ones (including many popular Android phones) usually don't.
On the other hand, as you say, MIFARE Classic supports capabilities beyond NFC/NDEF, but there are fully NFC-compliant tags that do so as well (e.g. MIFARE DESfire, which properly stacks encryption in an ISO 14443-4 compliant way).