test: refactor browser test utils

[hi-ogawa]

Description

Currently runBrowserTests utility is relying on json reporter output browser.json, which is causing a cryptic CI error in #6512. I rewrote this part to use assertions based on vitest state API.

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[sheremet-va]

We use the json reporter to also indirectly test it. I think if you see cryptic errors, you can change the assertions themselves. Tests already print the stdout and stderr when failed.

[hi-ogawa]

We use the json reporter to also indirectly test it.

Yeah, if that's important, we can keep it for runner.test.ts, but I don't think we need to do it from runBrowserTests itself. I reverted the second commit and how does it look?

[sheremet-va]

We use the json reporter to also indirectly test it.

Yeah, if that's important, we can keep it for runner.test.ts, but I don't think we need to do it from runBrowserTests itself. I reverted the second commit and how does it look?

Looks good to me 👍🏻