🌱 improve release-staging build

[cahillsf]

What this PR does / why we need it:
This reintroduces changes made in: #9536

Last time this change was merged it exposed use to a bug in the docker build server version we are using and the change was reverted. Details here: #9752

---

This change has been merged to prevent this bug from occurring: #9990

And we have also introduced an additional step in the release-staging target to ensure the images contain the expected binaries: #9932

/area release

[k8s-ci-robot]

Skipping CI for Draft Pull Request.
If you want CI signal for your change, please convert it to an actual PR.
You can still manually trigger a test run with /test all

[chrischdi]

/lgtm

[k8s-ci-robot]

LGTM label has been added.

Git tree hash: 552318c4409a2d289fa4977bc2393e6170822ddd

[sbueringer]

🤞
/approve

[k8s-ci-robot]

[APPROVALNOTIFIER] This PR is APPROVED

This pull-request has been approved by: sbueringer

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[cahillsf]

/cherry-pick release-1.5

[cahillsf]

/cherry-pick release-1.6

[k8s-infra-cherrypick-robot]

@cahillsf: #10095 failed to apply on top of branch "release-1.5":

Applying: improve release-staging build
Using index info to reconstruct a base tree...
M	Makefile
Falling back to patching base and 3-way merge...
Auto-merging Makefile
CONFLICT (content): Merge conflict in Makefile
error: Failed to merge in the changes.
hint: Use 'git am --show-current-patch=diff' to see the failed patch
Patch failed at 0001 improve release-staging build
When you have resolved this problem, run "git am --continue".
If you prefer to skip this patch, run "git am --skip" instead.
To restore the original branch and stop patching, run "git am --abort".

In response to this:

/cherry-pick release-1.5

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/test-infra repository.

[k8s-infra-cherrypick-robot]

@cahillsf: #10095 failed to apply on top of branch "release-1.6":

Applying: improve release-staging build
Using index info to reconstruct a base tree...
M	Makefile
Falling back to patching base and 3-way merge...
Auto-merging Makefile
CONFLICT (content): Merge conflict in Makefile
error: Failed to merge in the changes.
hint: Use 'git am --show-current-patch=diff' to see the failed patch
Patch failed at 0001 improve release-staging build
When you have resolved this problem, run "git am --continue".
If you prefer to skip this patch, run "git am --skip" instead.
To restore the original branch and stop patching, run "git am --abort".

In response to this:

/cherry-pick release-1.6

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/test-infra repository.