Missing new-notification icon indicator for "added to reading list" action

[stephendonner]

Description

Missing new-notification icon indicator for "added to reading list" action

According to #17959 this should be there (first testcase!)

Steps to Reproduce

1. install 1.44.73
2. launch Brave
3. load theverge.com
4. right-click on the tab
5. choose Add Tab to Reading List
6. look at the side panel toolbar icon

Actual result:

Expected result:

Reproduces how often:

100%

Brave version (brave://version info)

Brave	1.44.73 Chromium: 105.0.5195.68 (Official Build) beta (x86_64)
Revision	ad13e82529051bac6a0e65f455e6d7a1e5fd7938-refs/branch-heads/5195@{#903}
OS	macOS Version 13.0 (Build 22A5331f)

Version/Channel Information:

• Can you reproduce this issue with the current release?
• Can you reproduce this issue with the beta channel?
• Can you reproduce this issue with the nightly channel?

Other Additional Information:

• Does the issue resolve itself when disabling Brave Shields?
• Does the issue resolve itself when disabling Brave Rewards?
• Is the issue reproducible on the latest version of Chrome?

Miscellaneous Information:

/cc @rebron @aguscruiz @petemill @GeetaSarvadnya

[aguscruiz]

Added designs for this on Figma:
https://www.figma.com/file/xOdKFSr3P5AXSAbP2r8ung/Sidebar-Desktop?node-id=4272%3A70774

[rebron]

cc: @petemill