Google Maps says it's ‘Awahua Beach.
Correct. It's the beach where you dismount when you take the mules down the trail to Kalaupapa. With a 2000 ft. change in elevation, not many people would make the trek just to hang out at 'Awahua. So, yeah, 'Awahua is usually pretty deserted.
I believe the article you cited may be incorrect in stating that a permit is needed to visit 'Awahua. I don't think a permit is needed to descend the trail to "Awahua. At the time we went, my understanding was that a permit and tour group reservation was needed to proceed beyond the immediate area where the trail reached the beach area. When we were on the mules, we did pass a few people on the trail who didn't appear to be part of a tour group - but perhaps they had made arrangements to join a tour whenever it was they did reach the bottom.
DW had friends who visited Molokai regularly (they owned a condo at Ke Nani Kai), and the husband occasionally hiked down to 'Awahua and back up for exercise.