Subscribe block: the block becomes inactive after submitting the first email address

### Impacted plugin

Jetpack(on WordPress.com

### Quick summary

Enter one email in the subscribe block and the block greys itself out.

### Steps to reproduce

1) Create a site with a Business plan and add the subscribe block (free subscriptions)
2) Go to the site in an incognito browser
3) Go to the area where the subscribe block is located
4) Enter an email address and subscribe. Click to accept the message
5) Notice how the email subscribe box is now greyed out. You can't enter a new email.

This can be recreated reliably on a test site too with the same plan and theme.

Video

https://github.com/user-attachments/assets/5ca82205-f9c1-4b48-9f4a-b4607325ffd5

### Site owner impact

Fewer than 20% of the total website/platform users

### Severity

Major

### What other impact(s) does this issue have?

_No response_

### If a workaround is available, please outline it here.

_No response_

### Platform (Simple and/or Atomic)

_No response_

No source files, tests, or entry points are named. Reproduce the issue using the listed Business-plan and incognito-browser steps, then trace the subscribe block behavior after accepting the message; it is done when a submitted email no longer leaves the block greyed out and a new email can be entered.

Written by the indexing model from the issue text.