Setting up Stripe connection from FSE results in a 'Sorry, you are not allowed to access this page.' message

### Impacted plugin

Jetpack

### Steps to Reproduce

1. Create a new Jurassic Ninja site
2. Use Appearance -> Edit to open up FSE
3. Add a payment button
4. Click 'Connect stripe'
5. Go through the Stripe sign-up
6. In stripe click the button to connect
7. Get redirected back to the JN site

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

* Should be returned back to the template being edited
* Should see a message like 'Congrats! Your site is now connected to Stripe. You can now start accepting funds!'

### What actually happened

I was redirected back to an FSE looking url: https://identical-seahorse.jurassic.ninja/wp-admin/admin.php?stripe_connect_success=gutenberg&page=gutenberg-edit-site&postId=twentytwentytwo%2F%2Fhome&postType=wp_template but I had an error message saying 'Sorry, you are not allowed to access this page.', though the connection had worked.


### Other information

Video of it happening:
https://user-images.githubusercontent.com/93301/160138340-01fcbc4d-da95-4116-b1a4-e5eab8d272e3.mp4

This only happens when initiating the connection from FSE, not from within the post editor.

Note that I haven't been able to reproduce this on WordPress.com simple sites.

### Operating System

_No response_

### OS Version

_No response_

### Browser

Firefox

### Browser Version(s)

_No response_

Start by reproducing the listed FSE Stripe connection flow and inspect handling of the stripe_connect_success=gutenberg redirect to gutenberg-edit-site. Done means the user returns to the template being edited and sees a successful Stripe connection message instead of an access error.

Written by the indexing model from the issue text.