Backport removal of twine --skip-existing test [3.11] #959

Backport to 3.10: 💔 cherry-picking failed — conflicts found

❌ Failed to cleanly apply 3b820bc on top of patchback/backports/3.10/3b820bc42c9da015debebb822b11f1c181bf07dc/pr-959

Ensure you have a local repo clone of your fork. Unless you cloned it
from the upstream, this would be your origin remote.
Make sure you have an upstream repo added as a remote too. In these
instructions you'll refer to it by the name upstream. If you don't
have it, here's how you can add it:
```
$ git remote add upstream https://github.com/pulp/pulp_python.git
```

Ensure you have the latest copy of upstream and prepare a branch
that will hold the backported code:

$ git fetch upstream
$ git checkout -b patchback/backports/3.10/3b820bc42c9da015debebb822b11f1c181bf07dc/pr-959 upstream/3.10

Now, cherry-pick PR #959 contents into that branch:
```
$ git cherry-pick -x 3b820bc42c9da015debebb822b11f1c181bf07dc
```
If it'll yell at you with something like fatal: Commit 3b820bc42c9da015debebb822b11f1c181bf07dc is a merge but no -m option was given., add -m 1 as follows instead:
```
$ git cherry-pick -m1 -x 3b820bc42c9da015debebb822b11f1c181bf07dc
```
At this point, you'll probably encounter some merge conflicts. You must
resolve them in to preserve the patch from PR #959 as close to the
original as possible.

Push this branch to your fork on GitHub:

$ git push origin patchback/backports/3.10/3b820bc42c9da015debebb822b11f1c181bf07dc/pr-959

Create a PR, ensure that the CI is green. If it's not — update it so that
the tests and any other checks pass. This is it!
Now relax and wait for the maintainers to process your pull request
when they have some cycles to do reviews. Don't worry — they'll tell you if
any improvements are necessary when the time comes!