[Bug]: Notifications do not appear on the MetaMask badge; when a user toggles any of the notification settings on and off, there seems to be a pause in notification delivery

Nobody has claimed this yet.

### Describe the bug

1. Notifications appear only within the MetaMask tab or popup, but not on the MM badge - not sure if that's expected.
2. When a user toggles any of the notification settings on and off, there seems to be a pause in notification delivery. Notifications resume once the user navigates to Notifications -> Settings, they then appear exclusively within the MetaMask tab or popup.

### Expected behavior

-

### Screenshots/Recordings

Chrome:

https://github.com/user-attachments/assets/cdec85cd-626e-4558-8aa1-f12b45ed739d

Firefox:

https://github.com/user-attachments/assets/aaa93420-4f4a-4a92-86cf-5c27a1d68b1d

### Steps to reproduce

1. Go to Notifications -> Settings, disable / enbale any of the toggles for Chrome (for Firefox you don't need to ON/OFF toggles, notifications do not show up after enabling them)
2. Send a transaction (-s)
3. Notification doesn't appear
4. Go back to Notifications -> Settings
5. Notifications for the sent transactions displayed

### Error messages or log output

_No response_

### Detection stage

In production (default)

### Version

12.1.0

### Build type

None

### Browser

Chrome, Firefox

### Operating system

MacOS

### Hardware wallet

_No response_

### Additional context

_No response_

### Severity

_No response_

Start by reproducing the Chrome and Firefox steps in Notifications → Settings, including sending a transaction after changing notification toggles. Compare badge delivery with notifications shown in the MetaMask tab or popup, and consider the issue resolved when notifications appear without reopening Settings after a toggle change.

Written by the indexing model from the issue text.