Does the USB-C Alt Mode specification require support of the USB 3 protocol?
From what I can read from the USB Type C Specification release 1.0 and the USB Power Delivery Specification, I'm feeling that I might go as far as to say that if you don't need the USB functionality in your device, then you might not need to provide any USB device circuitry at all. 

The USB-C spec chapter 5.1.1 says that the alternate mode is selected by USB Power Delivery ("PD") signaling. And taking a look at the USB Power Delivery specification chapter 5.8.3 says that in USB-C the PD messages are sent with baseband BMC signaling on the CC line, not in the USB data lines. Furthermore, 5.8.1 says that the bitrate is 300kbps. This would hint to the direction that more simple circuitry than a USB device or host port would be necessary to carry the PD messages needed to select the alternative mode.

If you need USB, consider this snippet from tomshardware:


  USB Type-C is compatible with previous iterations of USB, at least in
  that it supports all prior protocols from USB 2.0 onward, including
  the drive stack, capabilities and infrastructure.


Based on this, for any USB functionality, I would say that it's perfectly ok even to use ancient 12 mbps USB circuitry, if it's fast enough for your design. 12 Mbps 'Full speed' and even 1.5 Mbps 'slow speed' are covered by the USB 2.0 specification.



[Edit] Please see the comments below for discussion about the "USB Billboard Device" interface which may be required, seemingly for the purpose of showing "This device doesn't support USB" types of messages via USB. If such an interface is required, then even low-speed (1.5 Mbps) should be enough. Further clarification from more experts is welcome.