refactor: Switch write api usage to use AppendRowsStream helper

[TrevorBergeron]

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Fixes #<issue_number_goes_here> 🦕

[chalmerlowe]

📢 Migration Notice: 📢 This library is moving to the google-cloud-python monorepo soon.

We kept this PR open due to recent activity. We would like to finalize this PR so it can be merged if it is critical.

If we don't hear from the PR author, we will close this PR in the next few days. The PR can then be re-opened in the monorepo once the migration is complete and work can continue there.

[tswast]

Closing. Let's move this over to https://github.com/googleapis/google-cloud-python once googleapis/google-cloud-python#16493 merges.