CI: Migrate Python 3.10 testing to Posix GHA/Azure Pipelines
- [ ] closes #43890
- [ ] tests added / passed
- [ ] Ensure all linting tests pass, see [here](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit) for how to run them
- [ ] whatsnew entry

Can we backport this? It would be nice to keep CI as shared as possible between the release and main branches before the actual release is cut. 
This is also pretty low risk given that it only changes CI and not code. yep ok to merge and backport  thanks @lithomas1  @meeseeksdev backport 1.4.x Owee, I'm MrMeeseeks, Look at me.

There seem to be a conflict, please backport manually. Here are approximate instructions:

1. Checkout backport branch and update it.

```
git checkout 1.4.x
git pull
```

2. Cherry pick the first parent branch of the this PR on top of the older branch:
```
git cherry-pick -m1 d437902f46acbff4a03d748b30620bc75fa5ea1f
```

3. You will likely have some merge/cherry-pick conflict here, fix them and commit:

```
git commit -am 'Backport PR #45120: CI: Migrate Python 3.10 testing to Posix GHA/Azure Pipelines'
```

4. Push to a named branch:

```
git push YOURFORK 1.4.x:auto-backport-of-pr-45120-on-1.4.x
```

5. Create a PR against branch 1.4.x, I would have named this PR:

> "Backport PR #45120 on branch 1.4.x (CI: Migrate Python 3.10 testing to Posix GHA/Azure Pipelines)"

And apply the correct labels and milestones.

Congratulations — you did some good work! Hopefully your backport PR will be tested by the continuous integration and merged soon!

Remember to remove the `Still Needs Manual Backport` label once the PR gets merged.

If these instructions are inaccurate, feel free to [suggest an improvement](https://github.com/MeeseeksBox/MeeseeksDev).
                 Owee, I'm MrMeeseeks, Look at me.

There seem to be a conflict, please backport manually. Here are approximate instructions:

1. Checkout backport branch and update it.

```
git checkout 1.4.x
git pull
```

2. Cherry pick the first parent branch of the this PR on top of the older branch:
```
git cherry-pick -m1 d437902f46acbff4a03d748b30620bc75fa5ea1f
```

3. You will likely have some merge/cherry-pick conflict here, fix them and commit:

```
git commit -am 'Backport PR #45120: CI: Migrate Python 3.10 testing to Posix GHA/Azure Pipelines'
```

4. Push to a named branch:

```
git push YOURFORK 1.4.x:auto-backport-of-pr-45120-on-1.4.x
```

5. Create a PR against branch 1.4.x, I would have named this PR:

> "Backport PR #45120 on branch 1.4.x (CI: Migrate Python 3.10 testing to Posix GHA/Azure Pipelines)"

And apply the correct labels and milestones.

Congratulations — you did some good work! Hopefully your backport PR will be tested by the continuous integration and merged soon!

Remember to remove the `Still Needs Manual Backport` label once the PR gets merged.

If these instructions are inaccurate, feel free to [suggest an improvement](https://github.com/MeeseeksBox/MeeseeksDev).
                