Accessing service from an Alpine-based k8s pod is throwing a DNS Resolution error

I have pod A (it's actually the kube-scheduler pod) and pod B (a pod that has a REST API that will be invoked by pod A).

For this purpose, I created a ClusterIP service.

Now, when I exec into pod A to perform the API call to pod B, I get: curl: (6) Could not resolve host: my-svc.default.svc.cluster.local

I tried to follow the debug instructions mentioned here:

kubectl exec -i -t dnsutils -- nslookup my-svc.default
Server:         10.96.0.10
Address:        10.96.0.10#53

Name:   my-svc.default.svc.cluster.local
Address: 10.111.181.13

Also:

kubectl exec -i -t dnsutils -- nslookup kubernetes.default
Server:         10.96.0.10
Address:        10.96.0.10#53

Name:   kubernetes.default.svc.cluster.local
Address: 10.96.0.1

This seems to be working as expected. However, when I exec into pod A, I get:

kubectl exec -it kube-scheduler -n kube-system -- sh
/bin # nslookup kubernetes.default
Server:         8.8.8.8
Address:        8.8.8.8:53

** server can't find kubernetes.default: NXDOMAIN

** server can't find kubernetes.default: NXDOMAIN

Other debugging steps (inside pod A) include:

/bin # cat /etc/resolv.conf
nameserver 8.8.8.8
nameserver 172.30.0.1

And:

/bin # cat /etc/*-release
3.12.8
NAME="Alpine Linux"
ID=alpine
VERSION_ID=3.12.8
PRETTY_NAME="Alpine Linux v3.12"
HOME_URL="https://alpinelinux.org/"
BUG_REPORT_URL="https://bugs.alpinelinux.org/"

There are no useful logs from the coredns pods, either.

kubectl logs --namespace=kube-system -l k8s-app=kube-dns
.:53
[INFO] plugin/reload: Running configuration MD5 = db32ca3650231d74073ff4cf814959a7
CoreDNS-1.7.0
linux/amd64, go1.14.4, f59c03d
.:53
[INFO] plugin/reload: Running configuration MD5 = db32ca3650231d74073ff4cf814959a7
CoreDNS-1.7.0
linux/amd64, go1.14.4, f59c03d

From the documentation, it seems there is a known issue with Alpine and DNS resolution (even though the version I have is greater than the version they mentioned).

Is there a workaround this to enable accessing the service properly from the Alpine pod?

Edit providing pod A manifest:

apiVersion: v1
kind: Pod
metadata:
  creationTimestamp: null
  labels:
    component: kube-scheduler
    tier: control-plane
  name: kube-scheduler
  namespace: kube-system
spec:
  containers:
  - command:
    - kube-scheduler
    - --authentication-kubeconfig=/etc/kubernetes/scheduler.conf
    - --authorization-kubeconfig=/etc/kubernetes/scheduler.conf
    - --bind-address=127.0.0.1
    - --config=/etc/kubernetes/sched-cs.yaml
    - --port=0
    image: localhost:5000/scheduler-plugins/kube-scheduler:latest
    imagePullPolicy: IfNotPresent
    livenessProbe:
      failureThreshold: 8
      httpGet:
        host: 127.0.0.1
        path: /healthz
        port: 10259
        scheme: HTTPS
      initialDelaySeconds: 10
      periodSeconds: 10
      timeoutSeconds: 15
    name: kube-scheduler
    resources:
      requests:
        cpu: 100m
    startupProbe:
      failureThreshold: 24
      httpGet:
        host: 127.0.0.1
        path: /healthz
        port: 10259
        scheme: HTTPS
      initialDelaySeconds: 10
      periodSeconds: 10
      timeoutSeconds: 15
    volumeMounts:
    - mountPath: /etc/kubernetes/scheduler.conf
      name: kubeconfig
      readOnly: true
    - mountPath: /etc/kubernetes/sched-cs.yaml
      name: sched-cs
      readOnly: true
  hostNetwork: true
  priorityClassName: system-node-critical
  volumes:
  - hostPath:
      path: /etc/kubernetes/scheduler.conf
      type: FileOrCreate
    name: kubeconfig
  - hostPath:
      path: /etc/kubernetes/sched-cs.yaml
      type: FileOrCreate
    name: sched-cs
status: {}

Edit 2: Adding the following lines manually to /etc/resolv.conf of Pod A allows me to perform the curl request successfully.

nameserver 10.96.0.10
search default.svc.cluster.local svc.cluster.local cluster.local
options ndots:5

Wouldn't there be a cleaner/less manual way to achieve the same result?