Seems to be introduced in #85985. Failed 6 times in the last week. It also fails on my local machine.

**Build scan:**
https://gradle-enterprise.elastic.co/s/ovz3v34txctg2/tests/:modules:data-streams:test/org.elasticsearch.datastreams.MetadataIndexTemplateServiceTests/testValidateTsdbDataStreamsReferringTsdbTemplate

**Reproduction line:**
`./gradlew ':modules:data-streams:test' --tests "org.elasticsearch.datastreams.MetadataIndexTemplateServiceTests.testValidateTsdbDataStreamsReferringTsdbTemplate" -Dtests.seed=7ECFBE4DFC807230 -Dbuild.snapshot=false -Dtests.jvm.argline="-Dbuild.snapshot=false" -Dtests.locale=es-BO -Dtests.timezone=SystemV/PST8PDT -Druntime.java=17`

**Applicable branches:**
master

**Reproduces locally?:**
Yes

**Failure history:**
https://gradle-enterprise.elastic.co/scans/tests?tests.container=org.elasticsearch.datastreams.MetadataIndexTemplateServiceTests&tests.test=testValidateTsdbDataStreamsReferringTsdbTemplate

**Failure excerpt:**
```
java.lang.AssertionError: expected [index.time_series.end_time] to be private but it was not

  at __randomizedtesting.SeedInfo.seed([7ECFBE4DFC807230:CB9BEC8C04D43A85]:0)
  at org.elasticsearch.cluster.metadata.MetadataCreateIndexService.validatePrivateSettingsNotExplicitlySet(MetadataCreateIndexService.java:1285)
  at org.elasticsearch.cluster.metadata.MetadataCreateIndexService.getIndexSettingsValidationErrors(MetadataCreateIndexService.java:1275)
  at org.elasticsearch.cluster.metadata.MetadataIndexTemplateService.validate(MetadataIndexTemplateService.java:1631)
  at org.elasticsearch.cluster.metadata.MetadataIndexTemplateService.validate(MetadataIndexTemplateService.java:1549)
  at org.elasticsearch.cluster.metadata.MetadataIndexTemplateService.validate(MetadataIndexTemplateService.java:1544)
  at org.elasticsearch.cluster.metadata.MetadataIndexTemplateService.validateIndexTemplateV2(MetadataIndexTemplateService.java:666)
  at org.elasticsearch.cluster.metadata.MetadataIndexTemplateService.addIndexTemplateV2(MetadataIndexTemplateService.java:617)
  at org.elasticsearch.datastreams.MetadataIndexTemplateServiceTests.testValidateTsdbDataStreamsReferringTsdbTemplate(MetadataIndexTemplateServiceTests.java:62)
  at jdk.internal.reflect.NativeMethodAccessorImpl.invoke0(NativeMethodAccessorImpl.java:-2)
  at jdk.internal.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:77)
  at jdk.internal.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
  at java.lang.reflect.Method.invoke(Method.java:568)
  at com.carrotsearch.randomizedtesting.RandomizedRunner.invoke(RandomizedRunner.java:1758)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$8.evaluate(RandomizedRunner.java:946)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$9.evaluate(RandomizedRunner.java:982)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$10.evaluate(RandomizedRunner.java:996)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at org.apache.lucene.tests.util.TestRuleSetupTeardownChained$1.evaluate(TestRuleSetupTeardownChained.java:44)
  at org.apache.lucene.tests.util.AbstractBeforeAfterRule$1.evaluate(AbstractBeforeAfterRule.java:43)
  at org.apache.lucene.tests.util.TestRuleThreadAndTestName$1.evaluate(TestRuleThreadAndTestName.java:45)
  at org.apache.lucene.tests.util.TestRuleIgnoreAfterMaxFailures$1.evaluate(TestRuleIgnoreAfterMaxFailures.java:60)
  at org.apache.lucene.tests.util.TestRuleMarkFailure$1.evaluate(TestRuleMarkFailure.java:44)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at com.carrotsearch.randomizedtesting.ThreadLeakControl$StatementRunner.run(ThreadLeakControl.java:375)
  at com.carrotsearch.randomizedtesting.ThreadLeakControl.forkTimeoutingTask(ThreadLeakControl.java:824)
  at com.carrotsearch.randomizedtesting.ThreadLeakControl$3.evaluate(ThreadLeakControl.java:475)
  at com.carrotsearch.randomizedtesting.RandomizedRunner.runSingleTest(RandomizedRunner.java:955)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$5.evaluate(RandomizedRunner.java:840)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$6.evaluate(RandomizedRunner.java:891)
  at com.carrotsearch.randomizedtesting.RandomizedRunner$7.evaluate(RandomizedRunner.java:902)
  at org.apache.lucene.tests.util.AbstractBeforeAfterRule$1.evaluate(AbstractBeforeAfterRule.java:43)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at org.apache.lucene.tests.util.TestRuleStoreClassName$1.evaluate(TestRuleStoreClassName.java:38)
  at com.carrotsearch.randomizedtesting.rules.NoShadowingOrOverridesOnMethodsRule$1.evaluate(NoShadowingOrOverridesOnMethodsRule.java:40)
  at com.carrotsearch.randomizedtesting.rules.NoShadowingOrOverridesOnMethodsRule$1.evaluate(NoShadowingOrOverridesOnMethodsRule.java:40)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at org.apache.lucene.tests.util.TestRuleAssertionsRequired$1.evaluate(TestRuleAssertionsRequired.java:53)
  at org.apache.lucene.tests.util.AbstractBeforeAfterRule$1.evaluate(AbstractBeforeAfterRule.java:43)
  at org.apache.lucene.tests.util.TestRuleMarkFailure$1.evaluate(TestRuleMarkFailure.java:44)
  at org.apache.lucene.tests.util.TestRuleIgnoreAfterMaxFailures$1.evaluate(TestRuleIgnoreAfterMaxFailures.java:60)
  at org.apache.lucene.tests.util.TestRuleIgnoreTestSuites$1.evaluate(TestRuleIgnoreTestSuites.java:47)
  at com.carrotsearch.randomizedtesting.rules.StatementAdapter.evaluate(StatementAdapter.java:36)
  at com.carrotsearch.randomizedtesting.ThreadLeakControl$StatementRunner.run(ThreadLeakControl.java:375)
  at com.carrotsearch.randomizedtesting.ThreadLeakControl.lambda$forkTimeoutingTask$0(ThreadLeakControl.java:831)
  at java.lang.Thread.run(Thread.java:833)

```
modules/data-streams/src/test/java/org/elasticsearch/datastreams/MetadataIndexTemplateServiceTests.java
@AwaitsFix(bugUrl = "https://github.com/elastic/elasticsearch/issues/86333")
    public void testValidateTsdbDataStreamsReferringTsdbTemplate() throws Exception {
        var state = ClusterState.EMPTY_STATE;
        final var service = getMetadataIndexTemplateService();
        var template = new ComposableIndexTemplate(
            Collections.singletonList("logs-*-*"),
            new Template(builder().put("index.routing_path", "uid").build(), new CompressedXContent(generateTsdbMapping()), null),
            null,
            100L,
            null,
            null,
            new ComposableIndexTemplate.DataStreamTemplate(false, false),
            null
        );
        state = service.addIndexTemplateV2(state, false, "logs", template);
        var now = Instant.now();
        var mBuilder = Metadata.builder(state.getMetadata());
        DataStreamTestHelper.getClusterStateWithDataStream(
            mBuilder,
            "unreferenced",
            List.of(Tuple.tuple(now.minus(2, ChronoUnit.HOURS), now))
        );
        DataStreamTestHelper.getClusterStateWithDataStream(
            mBuilder,
            "logs-mysql-default",
            List.of(Tuple.tuple(now.minus(2, ChronoUnit.HOURS), now))
        );