Adding an updated solution for the healthcheck approach.
Simple snippet:
healthcheck:
  test: out=$$(mysqladmin ping -h localhost -P 3306 -u foo --password=bar 2>&1); echo $$out | grep 'mysqld is alive' || { echo $$out; exit 1; }
Explanation:
Since mysqladmin ping returns false positives (especially for wrong password), I'm saving the output to a temporary variable, then using grep to find the expected output (mysqld is alive). If found it will return the 0 error code. In case it's not found, I'm printing the whole message, and returning the 1 error code.
Extended snippet:
version: "3.8"
services:
  db:
    image: linuxserver/mariadb
    environment:
      - FILE__MYSQL_ROOT_PASSWORD=/run/secrets/mysql_root_password
      - FILE__MYSQL_PASSWORD=/run/secrets/mysql_password
    secrets:
      - mysql_root_password
      - mysql_password
    healthcheck:
      test: out=$$(mysqladmin ping -h localhost -P 3306 -u root --password=$$(cat $${FILE__MYSQL_ROOT_PASSWORD}) 2>&1); echo $$out | grep 'mysqld is alive' || { echo $$out; exit 1; }
secrets:
  mysql_root_password:
    file: ${SECRETSDIR}/mysql_root_password
  mysql_password:
    file: ${SECRETSDIR}/mysql_password
Explanation: I'm using docker secrets instead of env variables (but this can be achieved with regular env vars as well). The use of $$ is for literal $ sign which is stripped when passed to the container.
Output from docker inspect --format "{{json .State.Health }}" db | jq on various occasions:
Everything alright:
{
  "Status": "healthy",
  "FailingStreak": 0,
  "Log": [
    {
    {
      "Start": "2020-07-20T01:03:02.326287492+03:00",
      "End": "2020-07-20T01:03:02.915911035+03:00",
      "ExitCode": 0,
      "Output": "mysqld is alive\n"
    }
  ]
}
DB is not up (yet):
{
  "Status": "starting",
  "FailingStreak": 1,
  "Log": [
    {
      "Start": "2020-07-20T01:02:58.816483336+03:00",
      "End": "2020-07-20T01:02:59.401765146+03:00",
      "ExitCode": 1,
      "Output": "\u0007mysqladmin: connect to server at 'localhost' failed error: 'Can't connect to local MySQL server through socket '/var/run/mysqld/mysqld.sock' (2 \"No such file or directory\")' Check that mysqld is running and that the socket: '/var/run/mysqld/mysqld.sock' exists!\n"
    }
  ]
}
Wrong password:
{
  "Status": "unhealthy",
  "FailingStreak": 13,
  "Log": [
    {
      "Start": "2020-07-20T00:56:34.303714097+03:00",
      "End": "2020-07-20T00:56:34.845972979+03:00",
      "ExitCode": 1,
      "Output": "\u0007mysqladmin: connect to server at 'localhost' failed error: 'Access denied for user 'root'@'localhost' (using password: YES)'\n"
    }
  ]
}