[test] Refactor most Pickers test to async

[LukasTy]

I suspect that the fake clock configuration might be leaking outside the intended scope and producing flaky unit tests (i.e. 1, 2, 3).

This PR aims to ensure that the timers mocking is reset after each describe block.
After more testing I concluded that manual clock restoring doesn't help at all as everything is already done.

I'm exploring the next best thing—avoid faking clock whenever possible and where needed, ensure that only Date is mocked to prevent running into cases where faked timeouts are bleeding into an async test.

The current solution in a nutshell:

• Add @testing-library/user-event to Pickers and test packages dependencies to avoid the need of passing click and keyboard function down to utility functions
  • if you think that more verbose tests are a good compromise to avoid this dependency, I can refactor it.
    But we are already bringing in this dependency in test/performance-charts package... 🤷
• Remove clock: 'fake' whenever possible
• Use userEvent.click and userEvent.keyboard instead of the sync fireEvent utility

[mui-bot]

Deploy preview: https://deploy-preview-14583--material-ui-x.netlify.app/

Generated by 🚫 dangerJS against 8e7281f

[LukasTy]

This is generated after running pnpm start

[JCQuintas]

Btw, while working on #14508 I noticed there are two instances that makes the tests somewhat unstable, these clocks, and anything that changes moment.locale

[cherniavskii]

I don't understand how this works, but I don't want to block this PR if it reduces the flakiness 😅

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[LukasTy]

I'm parking this PR until we move away from mocha. (#14508)
it appears to be the main problem in ensuring the test stability.
The configuration seems to be leaking outside to other tests and causing unexpected test failures. 🤷

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Hopefully, #14927 can reduce the existing test:unit pipeline flakiness in the meantime. 🤞

[github-actions]

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