Add MachineHealthCheck for control plane to private cluster e2e template

[nojnhuh]

What type of PR is this?
/kind flake

What this PR does / why we need it:

The private cluster e2e test seems to be hitting #5687. This PR ports the fix to most of the templates in #5696 to the private cluster template.

Which issue(s) this PR fixes (optional, in fixes #<issue number>(, fixes #<issue_number>, ...) format, will close the issue(s) when PR gets merged):
Fixes #

Special notes for your reviewer:

TODOs:

• squashed commits
• includes documentation
• adds unit tests
• cherry-pick candidate

Release note:

NONE

[k8s-ci-robot]

[APPROVALNOTIFIER] This PR is NOT APPROVED

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[nojnhuh]

/test pull-cluster-api-provider-azure-e2e-optional

[nojnhuh]

For better CI signal:

/cherry-pick release-1.20
/cherry-pick release-1.19

[k8s-infra-cherrypick-robot]

@nojnhuh: once the present PR merges, I will cherry-pick it on top of release-1.19, release-1.20 in new PRs and assign them to you.

In response to this:

For better CI signal:

/cherry-pick release-1.20
/cherry-pick release-1.19

Instructions for interacting with me using PR comments are available here. If you have questions or suggestions related to my behavior, please file an issue against the kubernetes-sigs/prow repository.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 53.06%. Comparing base (316370d) to head (aba6778).
⚠️ Report is 2 commits behind head on main.

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #5801   +/-   ##
=======================================
  Coverage   53.06%   53.06%           
=======================================
  Files         279      279           
  Lines       29677    29677           
=======================================
  Hits        15749    15749           
  Misses      13112    13112           
  Partials      816      816           

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[k8s-ci-robot]

@nojnhuh: The following test failed, say /retest to rerun all failed tests or /retest-required to rerun all mandatory failed tests:

Test name	Commit	Details	Required	Rerun command
pull-cluster-api-provider-azure-e2e	aba6778	link	true	/test pull-cluster-api-provider-azure-e2e

Full PR test history. Your PR dashboard. Please help us cut down on flakes by linking to an open issue when you hit one in your PR.

Instructions for interacting with me using PR comments are available here. If you have questions or suggestions related to my behavior, please file an issue against the kubernetes-sigs/prow repository. I understand the commands that are listed here.

[nojnhuh]

/test pull-cluster-api-provider-azure-e2e-optional

[nojnhuh]

/test pull-cluster-api-provider-azure-e2e-optional

[nojnhuh]

Testing kubernetes/test-infra#35109:

/test pull-cluster-api-provider-azure-load-test-custom-builds
/test pull-cluster-api-provider-azure-load-test-dra-custom-builds

[nojnhuh]

/test pull-cluster-api-provider-azure-load-test-dra-custom-builds

[nojnhuh]

/test pull-cluster-api-provider-azure-load-test-dra-custom-builds