Inbox: Remove woo plugin check for eligibility checker

[itsmeichigo]

Closes: #13322

Description

Previously in the eligibility checker for Inbox, we were checking version for the Woo plugin assuming there would be a change to filter inbox.

In #13214 we enabled the Inbox feature flag but didn't remove the Woo plugin check. This check is now redundant, so this PR removes it so that the eligibility check can be synchronous.

Also, I added a small improvement to support authenticating the admin page for self-hosted sites authenticated without WPCom.

Testing steps

• Confirm that the entry points to Inbox on both the dashboard and hub menu still work as before.
• Log out and log in again to a self-hosted store using application password.
• Enable the inbox card on the dashboard screen or navigate to the inbox list from the hub menu.
• Tap on any CTA that opens the admin page. Confirm that the page is authenticated automatically.

Screenshots

Hub menu	Dashboard

---

• 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 by scanning the QR code below to install the corresponding build.

	App Name	WooCommerce iOS
	Build Number	pr13325-a8b3076
	Version	19.5
	Bundle ID	com.automattic.alpha.woocommerce
	Commit	a8b3076
	App Center Build	WooCommerce - Prototype Builds #9997

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

[selanthiraiyan]

Nice improvement. This works as described and code LGTM. 🚀

I think the PR description should be updated to close #13322 instead of #13181.