Remove the app's uses of WordPressAuthenticator APIs

[crazytonyli]

Note

I recommend reviewing this PR commit by commit.

Description

The WordPressAuthenticator library is being removed through a stacked chain of PRs.

This PR removes the app's remaining uses of the library's APIs, without deleting any library code.

[dangermattic]

	1 Warning
⚠️	This PR is assigned to the milestone 27.0. This milestone is due in less than 4 days. Please make sure to get it merged by then or assign it to a milestone with a later deadline.

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[wpmobilebot]

📲 You can test the changes from this Pull Request in Jetpack by scanning the QR code below to install the corresponding build.

	App Name	Jetpack
	Configuration	Release-Alpha
	Build Number	32618
	Version	PR #25644
	Bundle ID	com.jetpack.alpha
	Commit	02e5cf5
	Installation URL	1b3fksfcm6ub0

Automatticians: You can use our internal self-serve MC tool to give yourself access to those builds if needed.

[wpmobilebot]

📲 You can test the changes from this Pull Request in WordPress by scanning the QR code below to install the corresponding build.

	App Name	WordPress
	Configuration	Release-Alpha
	Build Number	32618
	Version	PR #25644
	Bundle ID	org.wordpress.alpha
	Commit	02e5cf5
	Installation URL	3v1pnb9c95h1o

Automatticians: You can use our internal self-serve MC tool to give yourself access to those builds if needed.

[wpmobilebot]

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