There are already some efforts in that direction. You can check out r2dbc and the many vert.x sql clients. While they're not really jdbc, they're fully async like you described.
I was investigating this lately, and as you say, they are "efforts", and while laudable, I wouldn't go so far as to call them solutions. There are open issues/caveats as a result of the async model.
I do think there is something great about being able to continue to leverage existing JDBC-related tooling, given that it already exists.
I do think there is something great about being able to continue to leverage existing JDBC-related tooling, given that it already exists.