### Description

As we are about to start leveraging `Weight#count` to compute total hit counts (https://github.com/elastic/elasticsearch/pull/84778) and aggregation counts (https://github.com/elastic/elasticsearch/pull/81322), we should start instrumenting it so that we can get information about time spent in `Weight#count`.
server/src/test/java/org/elasticsearch/search/profile/query/QueryProfilerTests.java
assertThat(breakdown.get(QueryTimingType.SCORE.toString()), equalTo(0L));
        assertThat(breakdown.get(QueryTimingType.MATCH.toString()), greaterThan(0L));
        assertThat(breakdown.get(QueryTimingType.CREATE_WEIGHT.toString() + "_count"), greaterThan(0L));
        assertThat(breakdown.get(QueryTimingType.BUILD_SCORER.toString() + "_count"), greaterThan(0L));
        assertThat(breakdown.get(QueryTimingType.NEXT_DOC.toString() + "_count"), greaterThan(0L));
        assertThat(breakdown.get(QueryTimingType.ADVANCE.toString() + "_count"), equalTo(0L));
        assertThat(breakdown.get(QueryTimingType.SCORE.toString() + "_count"), equalTo(0L));
        assertThat(breakdown.get(QueryTimingType.MATCH.toString() + "_count"), greaterThan(0L));
        long rewriteTime = profiler.getRewriteTime();
        assertThat(rewriteTime, greaterThan(0L));
    }
    public void testCollector() throws IOException {
        TotalHitCountCollector collector = new TotalHitCountCollector();
        ProfileCollector profileCollector = new ProfileCollector(collector);
        assertEquals(0, profileCollector.getTime());
        final LeafCollector leafCollector = profileCollector.getLeafCollector(reader.leaves().get(0));
        assertThat(profileCollector.getTime(), greaterThan(0L));
        long time = profileCollector.getTime();
        leafCollector.setScorer(null);
        assertThat(profileCollector.getTime(), greaterThan(time));
        time = profileCollector.getTime();
        leafCollector.collect(0);
        assertThat(profileCollector.getTime(), greaterThan(time));
    }
    private static class DummyQuery extends Query {
        @Override
        public String toString(String field) {
            return getClass().getSimpleName();
        }
        @Override