There's a little buglet in #85387 (which was the PR that closed #84046), specifically, it doesn't account for the possibility that there's both a default/request **and** and a final pipeline, and that they can both touch the `@timestamp` field.

`IngestService#postIngest` was added in #85387, and it's intended that the logic there only runs **once**, after all the pipelines associated with the request have been executed. But as it's currently written, the logic actually runs after each individual pipeline runs.

The consequence of that is that if there's a default/request and a final pipeline in use for a document, and the final pipeline changes the timestamp field, the `rawTimestamp` recorded for the `IndexRequest` associated with the document will mistakenly use the value it sees after the default/request pipeline runs, and **not** the most correct one (from after when the final pipeline runs).

/cc @martijnvg 
server/src/test/java/org/elasticsearch/ingest/IngestServiceTests.java
DiscoveryNode node1 = new DiscoveryNode("_node_id1", buildNewFakeTransportAddress(), Map.of(), Set.of(), Version.CURRENT);
        DiscoveryNode node2 = new DiscoveryNode("_node_id2", buildNewFakeTransportAddress(), Map.of(), Set.of(), Version.CURRENT);
        Map<DiscoveryNode, IngestInfo> ingestInfos = new HashMap<>();
        ingestInfos.put(node1, new IngestInfo(List.of(new ProcessorInfo("set"), new ProcessorInfo("remove"))));
        ingestInfos.put(node2, new IngestInfo(List.of(new ProcessorInfo("set"))));
        ElasticsearchParseException e = expectThrows(
            ElasticsearchParseException.class,
            () -> ingestService.validatePipeline(ingestInfos, putRequest.getId(), pipelineConfig)
        );
        assertEquals("Processor type [remove] is not installed on node [" + node2 + "]", e.getMessage());
        assertEquals("remove", e.getMetadata("es.processor_type").get(0));
        assertEquals("tag2", e.getMetadata("es.processor_tag").get(0));
        var pipelineConfig2 = XContentHelper.convertToMap(putRequest.getSource(), false, putRequest.getXContentType()).v2();
        ingestInfos.put(node2, new IngestInfo(List.of(new ProcessorInfo("set"), new ProcessorInfo("remove"))));
        ingestService.validatePipeline(ingestInfos, putRequest.getId(), pipelineConfig2);
    }
    public void testExecuteIndexPipelineExistsButFailedParsing() {
        IngestService ingestService = createWithProcessors(
            Map.of("mock", (factories, tag, description, config) -> new AbstractProcessor("mock", "description") {
                @Override
                public IngestDocument execute(IngestDocument ingestDocument) {
                    throw new IllegalStateException("error");
                }
                @Override
                public String getType() {
                    return null;
                }
            })
        );
        ClusterState clusterState = ClusterState.builder(new ClusterName("_name")).build();