Well on the topic of the proposal, “continental” does seem like a fine replacement for “international”. A separate “international” tag could then signify the capability to handle overseas passengers and cargo. This would cover for example isolated destinations that handle a very low volume but are nonetheless international airports.
“Aerodrome” is already used by major airports so it already has traction. Seems good to go forward with it.
As for “importance” and “flight_range” these seem unnecessary. The strength of their services, their importance and range of flights coming in are one in the same. An important airport is one that host long range flights and airports won’t have more services than they need for economic reasons.
As for private and public, these seem like the territory of “access” and “operator:type”.
For military, the current implementation seems to work well.
For aeronautical sports, I’d say adding “sport” to the aerodrome would work well. A regional and local aerodrome can host gliding, or ultralights or parachuting but you wouldn’t classify them as “gliding”. If it has limited infrastructure, it would be an “airfield” or “airstrip”.
Also, and I’m sorry if I’m being pedantic, but can we use “local” for aerodromes? It’s as if we had towns and hamlets but not villages or primary and residential roads but not secondaries and tertiaries.