Remove deprecated NeptuneLogger code

[carmocca]

What does this PR do?

This should have been removed last release, but was missed.

I hope I did this correctly. Please take a look, Neptune devs.

Unblocks #14715

Does your PR introduce any breaking changes? If yes, please list them.

Removes deprecated code.

Before submitting

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• Did you list all the breaking changes introduced by this pull request?
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PR review

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cc @Blaizzy

[awaelchli]

@Blaizzy Do you want to have a quick look here before we merge? 😃

[Blaizzy]

@awaelchli yes, I would like that.

[Blaizzy]

@awaelchli

I left a few comments and will also be calling in someone from engineering to double-check the test cases.

But outside of that, I think it looks good.

Next steps:
After the engineer gives the green light 🟢 , we can merge with no problem.

[Blaizzy]

@awaelchli, the review is done 👍

Any questions about the changes requested, please tag me and @shnela directly. We are here to help you with anything you might need.