fix(web): Fixes 12267 (https://github.com/firebase/flutterfire/issues/12267)

[xurc]

Description

The changes are based on the accepted PR (#12250), which fixes a similar bug.

Related Issues

fixes #12267

Checklist

• I read the Contributor Guide and followed the process outlined there for submitting PRs.
• My PR includes unit or integration tests for all changed/updated/fixed behaviors (See Contributor Guide).
• All existing and new tests are passing.
• I updated/added relevant documentation (doc comments with ///).
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• I signed the CLA.
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Breaking Change

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• Yes, this is a breaking change.
• No, this is not a breaking change.

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