A few things:
1. Yes, the test needs to be supervised. I forget which US airline has a system whereby the sample-taking is overseen by a medical professional via zoom. So there are ways to make it work, and I believe HI has indicated that arrangement would be okay.
2. Yes, there are CLIA-certified labs in Canada, but it does look like the ones we are all familiar with as patients in Alberta are not. The AB government lab that does the covid tests has a couple of "stickers" on its website re. certification/affiliation, but CLIA isn't one of them.
3. The HI Governor proclamation is drafted a bit clumsily. It should be read as having two separate requirements:
Beginning October 15, passengers who successfully complete a pre-travel test will not be subject to the state’s 14-day quarantine program. Currently approved is the processing by [1] laboratories that are licensed or certified by Clinical Laboratories Improvement Amendments (CLIA) of [2] specimens for nucleic acid amplification testing approved or authorized by the United States Food and Drug Administration, pursuant to an Emergency Use Authorization or other authorization for COVID-19 testing.
The emergency use authorization relates specifically to the type of test. The Abbott ID NOW test for example has that exact type of approval. So, 1. the lab must be CLIA-certified; and 2. the test that the lab administers must be a nucleic acid test that's basically FDA-approved.