SmallRye Health leaks MDC properties between health checks

[xstefank]

Describe the bug

Initially reported in #41711.

MDC properties are leaked between different HealthChecks. We probably need to configure some stickiness for the MDC in the health check.

It was probably caused by #37352.

Reproducer https://github.com/xstefank/quarkus-reproducers/tree/main/mdc-propagation.

Expected behavior

Same as in Quarkus 3.6.1:

10:11:32 INFO  health-check=TestHC [or.ac.TestHC] (executor-thread-1) Health check
10:11:32 INFO  health-check=TestHC2 [or.ac.TestHC2] (executor-thread-1) Health check

Actual behavior

In Quarkus 3.12.1:

10:10:20 INFO  health-check=TestHC [or.ac.TestHC2] (vert.x-worker-thread-5) Health check
10:10:20 INFO  health-check=TestHC [or.ac.TestHC] (vert.x-worker-thread-6) Health check

or

10:10:19 INFO  health-check=TestHC2 [or.ac.TestHC2] (vert.x-worker-thread-6) Health check
10:10:19 INFO  health-check=TestHC2 [or.ac.TestHC] (vert.x-worker-thread-5) Health check

How to Reproduce?

Reproducer https://github.com/xstefank/quarkus-reproducers/tree/main/mdc-propagation.

Output of uname -a or ver

No response

Output of java -version

No response

Quarkus version or git rev

No response

Build tool (ie. output of mvnw --version or gradlew --version)

No response

Additional information

I'm already working on this.

[quarkus-bot]

/cc @Ladicek (smallrye), @jmartisk (health,smallrye), @phillip-kruger (smallrye), @radcortez (smallrye)