Tiered property taxation hasn't solved the the problems OP has described for Oahu; or Hawaii as a whole for that matter.

For example, Local NIMBYist restrictions, mostly those individuals that have resided there for a long time (and have the lower property tax rate you describe), have resulted in areas immediately adjacent to 40-story downtown developments being zoned exclusively for single-family homes; see Kaimuki, Upper Tantalus, Manoa, Punchbowl (to an extent), etc. Consequently, even homes built over 80 yrs ago with only ~1,300 sqft sell for about $1.1 million; https://www.zillow.com/homedetails/1001-Belser-St-Honolulu-H....

The majority of people employed in hospitality and trades that support the local tourist economy (electricians, masons, carpenters, etc.) are increasingly driven to the west side; where commutes to the city can easily be 1-2 hours normally (pre-COVID). This eventually becomes such a stressor that many of the people working these skilled trades leave for the mainland.

Prices shouldn't rise if you are allowed to build enough supply to meet demand. A lack of property interest isn't necessarily a bad thing for a place. Land isn't for speculation, it's for use.