Incorrect rate agg with custom _doc_count

[sorenlouv]

Versions affected: 7.15 and up

I'm seeing some odd results with the rate agg on documents that have a custom _doc_count and wondering if it's a bug, similar to #74617.

Reproduce problem

PUT rate-bug

POST rate-bug/_doc
{
  "_doc_count": 10,
  "@timestamp": "2021-09-14T22:33:37.477Z"
}

GET rate-bug/_search
{
  "size": 0,
  "aggs": {
    "timeseries": {
      "date_histogram": {
        "field": "@timestamp",
        "fixed_interval": "60s"
      },
      "aggs": {
        "throughput": {
          "rate": {
            "unit": "minute"
          }
        }
      }
    }
  }
}

Response
The search request will return:

{
  "aggregations" : {
    "timeseries" : {
      "buckets" : [
        {
          "key_as_string" : "2021-09-14T22:33:00.000Z",
          "key" : 1631658780000,
          "doc_count" : 10,
          "throughput" : {
            "value" : 1.0
          }
        }
      ]
    }
  }
}

Expectation
I'd expect that a bucket for 1 minute with 10 documents should have a rate of 10.

Actual
Rate is 1.0

[elasticmachine]

Pinging @elastic/es-analytics-geo (Team:Analytics)

[jpountz]

As per a discussion with @sqren, I updated the title and description to remove references to the histogram field since this bug can be reproduced with the rate agg and documents that have custom _doc_counts only.