Fix Site Switcher screen not dismissing after adding self-hosted site

[pmusolino]

Fixes #24091

The Site Switcher screen wasn't being dismissed automatically after successfully adding a self-hosted site. This PR adds proper handling of the WPSigninDidFinishNotification notification to ensure the screen is dismissed when authentication completes.

Testing Instructions

1. Log in with a wp.com account
2. Open Site Switcher
3. Add a self-hosted site
4. Verify that after successful authentication, the Site Switcher screen is dismissed automatically.

Screen recording

Regression Notes

1. Potential unintended areas of impact

2. What I did to test those areas of impact (or what existing automated tests I relied on)

3. What automated tests I added (or what prevented me from doing so)

PR submission checklist:

• [ ] I have completed the Regression Notes.
• [ ] I have considered adding unit tests for my changes.
• [ ] I have considered adding accessibility improvements for my changes.
• [ ] I have considered if this change warrants user-facing release notes and have added them to RELEASE-NOTES.txt if necessary.

Testing checklist:

• [ ] WordPress.com sites and self-hosted Jetpack sites.
• [ ] Portrait and landscape orientations.
• [ ] Light and dark modes.
• [ ] Fonts: Larger, smaller and bold text.
• [ ] High contrast.
• [ ] VoiceOver.
• [ ] Languages with large words or with letters/accents not frequently used in English.
• [ ] Right-to-left languages. (Even if translation isn’t complete, formatting should still respect the right-to-left layout)
• [ ] iPhone and iPad.
• [ ] Multi-tasking: Split view and Slide over. (iPad)

[dangermattic]

	1 Warning
:warning:	This PR is assigned to the milestone 26.0. The due date for this milestone has already passed. Please assign it to a milestone with a later deadline or check whether the release for this milestone has already been finished.

Generated by :no_entry_sign: Danger

[wpmobilebot]

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

	App Name	WordPress Alpha
	Configuration	Release-Alpha
	Build Number	pr24273-27fc699
	Version	25.8
	Bundle ID	org.wordpress.alpha
	Commit	27fc699eb6a82bae11a8fadc04cefd53447d58b6
	App Center Build	WPiOS - One-Offs #11856

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

[wpmobilebot]

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

	App Name	Jetpack Alpha
	Configuration	Release-Alpha
	Build Number	pr24273-27fc699
	Version	25.8
	Bundle ID	com.jetpack.alpha
	Commit	27fc699eb6a82bae11a8fadc04cefd53447d58b6
	App Center Build	jetpack-installable-builds #10879

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

[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	26857
	Version	PR #24273
	Bundle ID	com.jetpack.alpha
	Commit	7495f7b5e86ff164159b7448927aaf11be8275a3
	Installation URL	1gc2l0fdnr18g

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	26857
	Version	PR #24273
	Bundle ID	org.wordpress.alpha
	Commit	7495f7b5e86ff164159b7448927aaf11be8275a3
	Installation URL	0tvf8rpk2gc68

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

[wpmobilebot]

Version 25.9 has now entered code-freeze, so the milestone of this PR has been updated to 26.0.

[kean]

Will revisit this later with a potentially different solution that wouldn't require the observer at this level.