### Elasticsearch Version

many

### Installed Plugins

none

### Java Version

_bundled_

### OS Version

all

### Problem Description

The `filter_by_filter` optimization should kick in when it makes aggs faster. We have a switch to turn it off, but we advise folks not to because it usually does a good job. Well, we've found a case where the heuristics do a bad job of picking the implementation - specifically I've seen a case when the `_doc_count` field exists and we're slower. I'm not really sure if this is universal, but it could be. I'm trying to figure out if this is an all the time thing or a sometimes thing.

### Steps to Reproduce

index docs with `_doc_count`. run a terms agg on a field with a cardinality in the ball park of 500.

### Logs (if relevant)

_No response_
server/src/main/java/org/elasticsearch/search/aggregations/support/AggregationContext.java
private final SearchExecutionContext context;
        private final PreallocatedCircuitBreakerService preallocatedBreakerService;
        private final BigArrays bigArrays;
        private final Supplier<Query> topLevelQuery;
        private final AggregationProfiler profiler;
        private final MultiBucketConsumer multiBucketConsumer;
        private final Supplier<SubSearchContext> subSearchContextBuilder;
        private final BitsetFilterCache bitsetFilterCache;
        private final int randomSeed;
        private final LongSupplier relativeTimeInMillis;
        private final Supplier<Boolean> isCancelled;
        private final Function<Query, Query> filterQuery;
        private final boolean enableRewriteToFilterByFilter;
        private final boolean inSortOrderExecutionRequired;
        private final AnalysisRegistry analysisRegistry;
        private final List<Aggregator> releaseMe = new ArrayList<>();
        public ProductionAggregationContext(
            AnalysisRegistry analysisRegistry,
            SearchExecutionContext context,