Don’t prompt for DevTools installation for browser extension page

[alex-kinokon]

Fixes #11382.

Checklist:

• If this PR contains changes to the library itself (not necessary for e.g. docs updates), please include a changeset (see CONTRIBUTING.md)
• If this PR is a new feature, please reference an issue where a consensus about the design was reached (not necessary for small changes)
• Make sure all of the significant new logic is covered by tests

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[jerelmiller]

This change looks great to me and seems in line with what React DevTools do.

Thanks a bunch for the contribution!

[jerelmiller]

@alex-kinokon don't worry about the integration test failure. I'm seeing the same thing in another PR of mine. Once we figure that out for the main branch, I'd be happy to merge this.