The phase shifted noise is just opposite wave length noise. It is still noise that may damage your hearing. While it may reduce some, people get lulled into thinking ANR is hearing protection it is not. It is noise reduction (tricking the brain).
This is a myth. Although you're providing a somewhat accurate description of how the technology works, you're incorrect that it doesn't protect your hearing. A quick search on google will turn up plenty of sources from MDs and scientists that discusses the benefits of hearing protection with ANR. The 20-25 decibels of noise reduction from ANR is just as good as 20-25 decibels of noise reduction from earplugs.