Gitlab Runner cannot run `docker` commands in `docker` executor

This has been driving me nuts as I think I'm exactly following the documentation by GitLab for setting up DIND using socket in GitLab Runner so I can run docker commands in Gitlab CI job. But it keeps giving the following error -

Running with gitlab-runner 14.0.0 (3b6f852e)
  on Gitlab-HiddenLayer-Group-Runner GosSpAyH
Preparing the "kubernetes" executor
00:00
Using Kubernetes namespace: gitlab
Using Kubernetes executor with image docker:19.03.12 ...
Using attach strategy to execute scripts...
Preparing environment
00:07
Waiting for pod gitlab/runner-gosspayh-project-27874308-concurrent-0qkp2h to be running, status is Pending
Waiting for pod gitlab/runner-gosspayh-project-27874308-concurrent-0qkp2h to be running, status is Pending
    ContainersNotReady: "containers with unready status: [build helper]"
    ContainersNotReady: "containers with unready status: [build helper]"
Running on runner-gosspayh-project-27874308-concurrent-0qkp2h via gitlab-runner-gitlab-runner-6984874897-l9z5z...
Getting source from Git repository
00:02
Fetching changes with git depth set to 50...
Initialized empty Git repository in /builds/GosSpAyH/0/hiddenlayer/hl-tech-blog/.git/
Created fresh repository.
Checking out c48b6257 as master...
Skipping Git submodules setup
Executing "step_script" stage of the job script
00:00
$ docker info
Client:
 Debug Mode: false
Server:
ERROR: Cannot connect to the Docker daemon at unix:///var/run/docker.sock. Is the docker daemon running?
errors pretty printing info
Cleaning up file based variables
00:01
ERROR: Job failed: command terminated with exit code 1

Here is my toml configuration in values.yaml for GitLab Runner installation in my private Kubernetes cluster.

  config: |
    [[runners]]
      url = "https://gitlab.com/"
      executor = "docker"
      privileged = true
      [runners.docker]
        tls_verify = false
        image = "docker:19.03.12"
        privileged = true
        disable_cache = false
        volumes = ["/var/run/docker.sock:/var/run/docker.sock", "/cache"]
      [runners.cache]
        Insecure = false

and my .gitlab-ci.yml is the following -

image: docker:19.03.12

variables:
  DOCKER_DRIVER: overlay2

before_script:
    - docker info
    - echo "$CI_REGISTRY_USER | $CI_REGISTRY_PASSWORD | $CI_REGISTRY"
    - docker login -u "$CI_REGISTRY_USER" -p "$CI_REGISTRY_PASSWORD" $CI_REGISTRY

build:
  stage: build
  # Default branch leaves tag empty (= latest tag)
  # All other branches are tagged with the escaped branch name (commit ref slug)
  script:
    - |
      if [[ "$CI_COMMIT_BRANCH" == "$CI_DEFAULT_BRANCH" ]]; then
        tag=""
        echo "Running on default branch '$CI_DEFAULT_BRANCH': tag = 'latest'"
      else
        tag=":$CI_COMMIT_REF_SLUG"
        echo "Running on branch '$CI_COMMIT_BRANCH': tag = $tag"
      fi
    - docker build --pull -t "$CI_REGISTRY_IMAGE${tag}" -f deploy/Dockerfile .
    - docker push "$CI_REGISTRY_IMAGE${tag}"

Note: I'm intentionally leaving the docker-dind service from the .gitlab-ci.yaml file because the documentation says it is not needed.

Additional Information:

• Kubernetes Version: 1.20
• Gitlab Runner Version: 14.0.0

Running docker commands in CIs is a pretty common workflow and I'm starting to think if it's this difficult to setup, I may as well go back to old ways with using Jenkins.