How to only get the array nested subdocuments from dynamic depth level, with that document id and not having to iterate through it

Other MongoDB Topics
1 
[
  {
    "_id": "1",
    "name": "sanggiyo",
    "gender": "Male",
    "children": [
      {
        "name": "tono",
        "gender": "Male",
        "children": [
          {
            "name": "anton",
            "gender": "male",
            "children": [],
            "_id": "4"
          }
        ],
        "_id": "2"
      },
      {
        "name": "rin",
        "gender": "Female",
        "children": [],
        "_id": "3"
      }
    ],
    "createdAt": "2022-04-17T11:14:00.648Z",
    "updatedAt": "2022-04-17T11:14:16.277Z",
    "__v": 0
  }
]

lets say i have the id for the second nested object, or third level nested object (_id=2 or _id=3), or four level nested object (_id=4)

how to get only the document that is related to that id and not the whole document?
so far I have tried:

db.collection.aggregate([
  {
    $match: {
      $or: [
        {
          "_id": "1"
        },
        {
          "children._id": "1"
        },
        {
          "children.children._id": "1"
        }
      ]
    }
  },
  {
    "$replaceRoot": {
      "newRoot": {
        "$first": {
          "$filter": {
            "input": "$children",
            "as": "item",
            "cond": {
              $eq: [
                "$$item._id",
                "2"
              ]
            }
          }
        }
      }
    }
  }
])

I need every id can get their own document. so in this case i expect 4 different output from 4 different case.
case 1 with id=1, i can get all data
case 2 with id=2,

  {
    "_id": "2",
    "children": [
      {
        "_id": "4",
        "children": [],
        "gender": "male",
        "name": "anton"
      }
    ],
    "gender": "Male",
    "name": "tono"
  }

case 3 with id=3

  {
    "_id": "3",
    "children": [],
    "gender": "Female",
    "name": "rin"
  }

case 4 with id=4

          {
            "name": "anton",
            "gender": "male",
            "children": [],
            "_id": "4"
          }

this is mongo playground for testing purpose, this way i can only get case 2 and 3 work, it fail getting case 1 and 4