Jetpack Subscribe Block: Problems when changing from dev to prod

### Impacted plugin

Social

### Quick summary

I'm putting this under social as the Jetpack Subscribe [support doc](https://jetpack.com/support/subscriptions/import-subscribers/) is under Social on the website.

- When you build out a site in a development environment and enable the Jetpack Subscribe Block there are issues when you change the URL to the production URL
- The logo breaks on the pop-up and in WP.com

- In [WP.com](http://wp.com/), the user doesn't seem to actually have subscriber permissions as their subscription is attached to the development URL.


We understand this is connected to the Feed.

### Steps to reproduce

1. Create a development site and enable Jetpack
2. Add the Subscribe Block to the site
3. Add a few emails to the Subscribe Block for testing purposes
4. "Launch" the site
5. Click on Subscription Management
6. You'll see the development URL is there
7. If you then try to manage your subscription and change the settings, you aren'e able to as you're not technically subscribed to the production URL

### A clear and concise description of what you expected to happen.

I expect the URL to update for a seamless transition to the production URL automatically.
Here's a walkthrough from receiving the email to trying to manage preferences: https://d.pr/v/BK3qm4

If you go here: https://wordpress.com/read/subscriptions
and search for the blog, you can only find the development URL, I expect to see the production URL.

The big red flag here is that people aren't able to unsubscribe which I think may be a policy violation.

### What actually happened

_No response_

### Impact

All

### Available workarounds?

Yes, easy to implement

### Platform (Simple and/or Atomic)

_No response_

### Logs or notes

This occured on a WordPress.com site as well as a self-hosted site on Pressable.

- MishTalk.com (the dev URL was https://mishtalk-production.mystagingwebsite.com/)
- Zuvuya.Space (the dev URL was https://a8cteam5136.wpcomstaging.com/)

Start by reproducing the listed development-to-production steps with the Jetpack Subscribe Block, then inspect the Feed and Subscription Management integration referenced in the report. Done means the production URL replaces the development URL in WordPress.com subscriptions, subscriber permissions work there, and the logo remains intact in the popup and on WP.com.

Written by the indexing model from the issue text.