Jetpack Setup: Improve web connection flow for outdated plugin

[itsmeichigo]

Part of WOOMOB-965

Description

This PR updates the web connection flow:

• Replaces the site connection URL to {site_url}/wp-admin/admin.php?page=jetpack to avoid forcing users to update their plugin.
• Adds workaround to reuse the WPCom credentials when loading the authorization page.
• Restores site connection handling for cookie-nonce authenticated users that was removed in REST API: Integrate Jetpack setup flow after login #9113

Testing steps

Follow TC7 in pe5sF9-42S-p2 to confirm that Jetpack connection through the web flow works for outdated plugin.

Testing information

From my testing, I could not make the jetpack/v4/connection/data to not return isRegistered - so I use an up-to-date Jetpack plugin on my JN site and use Proxyman to update the response to not return isRegistered.

Screenshots

Simulator.Screen.Recording.-.iPhone.16.-.2025-08-11.at.15.52.04.mp4

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• I have considered if this change warrants user-facing release notes and have added them to RELEASE-NOTES.txt if necessary.

[wpmobilebot]

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

	App Name	WooCommerce iOS Prototype
	Build Number	pr16001-dc3dc48
	Version	23.0
	Bundle ID	com.automattic.alpha.woocommerce
	Commit	dc3dc48
	Installation URL	61ad8tp48m2r8

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

[hichamboushaba]

Nice work @itsmeichigo, the scenario works well for both:

• Connecting an existing store from the site picker.
• For Application Passwords where the user used a WebView for authorization.

But the flow fails for Application Passwords where native login was used, check my comment below.

Also, a very minor nit, we show validating for the connection step before the user handles the connection, normally it should be shown only after the connection was established, and we are validating it, notice the difference between Android and iOS:

Android	iOS
Screen_recording_20250813_161423.mp4	Simulator.Screen.Recording.-.iPhone.16.-.2025-08-13.at.15.58.03.mp4

As said above, this is very minor, so I wouldn't spend a lot of time on it.

[hichamboushaba]

This condition doesn't work as expected when the user is signed in using site credentials natively, as we'll get .wporg credentials in this case:

The same issue also occurs in tracking here and I missed during the review of the previous PR (sorry).

[itsmeichigo]

Thank you Hicham for spotting the issues! I addressed them in subsequent commits, this PR is ready for another look 🙏

[hichamboushaba]

Nice work @itsmeichigo,