Missing Hazelcast logs after change #15636
## Describe the bug
We use Hazelcast embedded in one of our projects. After upgrading to 1.13.0.Final version we get `"LogManager error of type WRITE_FAILURE: The delayed handler's queue was overrun and log record(s) were lost. Did you forget to configure logging?"` during server startup. All Hazelcast startup logs are lost. I suspect it is related to change https://github.com/quarkusio/quarkus/pull/15636.

Hazelcast is initialized in a producer method. I added some logging into this producer method to see if they are printed however those logs get lost as well.

PS: When I start the project in `dev` mode the logs are printed correctly.
diff --git a/independent-projects/bootstrap/runner/src/main/java/io/quarkus/bootstrap/logging/QuarkusDelayedHandler.java b/independent-projects/bootstrap/runner/src/main/java/io/quarkus/bootstrap/logging/QuarkusDelayedHandler.java
index b39f29a5d57..74883b44991 100644
--- a/independent-projects/bootstrap/runner/src/main/java/io/quarkus/bootstrap/logging/QuarkusDelayedHandler.java
+++ b/independent-projects/bootstrap/runner/src/main/java/io/quarkus/bootstrap/logging/QuarkusDelayedHandler.java
@@ -23,6 +23,7 @@
 import java.util.logging.ErrorManager;
 import java.util.logging.Formatter;
 import java.util.logging.Handler;
+import java.util.logging.Level;
 import java.util.logging.Logger;
 import org.jboss.logmanager.ExtHandler;
 import org.jboss.logmanager.ExtLogRecord;
@@ -66,6 +67,10 @@ protected void doPublish(final ExtLogRecord record) {
                     publishToNestedHandlers(record);
                     super.doPublish(record);
                 } else {
+                    // until the handler is fully activated, we drop anything below the info level
+                    if (record.getLevel().intValue() < Level.INFO.intValue()) {
+                        return;
+                    }
                     // Determine whether the queue was overrun
                     if (logRecords.size() >= queueLimit) {
                         reportError(
['independent-projects/bootstrap/runner/src/main/java/io/quarkus/bootstrap/logging/QuarkusDelayedHandler.java']