chore: rewrite UI tests using @vitest/browser

[sheremet-va]

Description

Closes #5703

Install in CI is much faster now 🙏🏻

Please don't delete this checklist! Before submitting the PR, please make sure you do the following:

• It's really useful if your PR references an issue where it is discussed ahead of time. If the feature is substantial or introduces breaking changes without a discussion, PR might be closed.
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Tests

• Run the tests with pnpm test:ci.

Documentation

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Changesets

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