Reapply "Add check_for_installed_private_headers_in_cmake_out" (#13485)

[swolchok]

This reverts commit f15982d (#13685).

[swolchok]

Stack from ghstack (oldest at bottom):

• Add error logging to scripts/print_exported_headers #13686
• -> Reapply "Add check_for_installed_private_headers_in_cmake_out" (#13485) #13880

[pytorch-bot]

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❌ 5 New Failures, 1 Cancelled Job, 1 Unrelated Failure

As of commit b3b6210 with merge base be65301 ():

NEW FAILURES - The following jobs have failed:

• Build Presets / windows (pybind) / build (gh)
  Process completed with exit code 1.
• Build Presets / windows (windows) / build (gh)
  Process completed with exit code 1.
• pull / unittest / macos / macos-job (gh)
  RuntimeError: Command bash /Users/ec2-user/runner/_work/_temp/exec_script failed with exit code 1
• pull / unittest-editable / macos / macos-job (gh)
  RuntimeError: Command bash /Users/ec2-user/runner/_work/_temp/exec_script failed with exit code 1
• trunk / unittest-release / macos / macos-job (gh)
  RuntimeError: Command bash /Users/ec2-user/runner/_work/_temp/exec_script failed with exit code 1

CANCELLED JOB - The following job was cancelled. Please retry:

• trunk / test-huggingface-transformers-coreml (qwen3-1.7b|xnnpack|--quantize) / macos-job (gh)
  ##[error]The operation was canceled.

BROKEN TRUNK - The following job failed but were present on the merge base:

👉 Rebase onto the `viable/strict` branch to avoid these failures

• pull / unittest-nxp-neutron / linux-job (gh) (trunk failure)
  test_split_group_convolution__applied_by_default

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