No, you cannot use modern Intel systems without the ME. me_cleaner will only remove parts of it. What happens exactly (whether it actually stops working or goes into some undocumented free-for-all debug mode) is unknown as there is no introspection into the ME. HAP bit is asking the ME nicely to disable itself, sometime after it booted the system.
https://twitter.com/rootkovska/status/939064351008395264
Purism routinely overstates their capabilities in this regard, claiming to "neutralize" the ME.
Also note that the ME is a hardware feature. Most efforts to remove/disable it focus on the ME firmware, which is loaded only some time after boot. Some ME function remains even if you completely zero out the firmware.
See Peter Stuge's 30C3 Talk "Hardening hardware and choosing a #goodBIOS", noting IPv6 packet sent over the network interface even then (around 17:18 mark).