Expose the default `HealthCheckUpdateHandler` #11766

Job	Run time
lint	13m 44s
site	9m 57s
flaky-tests	6m 55s
build-ubicloud-standard-8-jdk-21-snapshot-blockhound	25m 57s
build-macos-12-jdk-21	1h 5m 45s
build-windows-latest-jdk-21	1h 5m 36s
build-ubicloud-standard-8-jdk-8	15m 46s
build-ubicloud-standard-8-jdk-11	18m 28s
build-ubicloud-standard-8-jdk-17-leak	19m 35s
build-ubicloud-standard-8-jdk-17-min-java-11	17m 45s
build-ubicloud-standard-8-jdk-17-min-java-17-coverage	25m 5s
	4h 44m 33s

Provide feedback