Dynamically set extension toolbar icons fail to apply in new windows until navigation or opening another tab

Dynamically set extension toolbar icons fail to apply in new windows until navigation or opening another tab.

1. Dynamically set the icon to something different from what's in the manifest. For example, with MV2, `chrome.browserAction.setIcon({ path: "icons/new_icon.svg" })`
2. Observe the new icon was applied to all existing tabs
3. Open a new window
4. Observe that the new window has the default extension icon specified in the manifest
5. You can try dynamically setting the icon again but the new icon will not be applied to the new window now either
6. Either open a tab or navigate somewhere in the new window. You will then see the new icon in that window, without having to call `setIcon()` again.

In step 4, I expect to see the new icon in a newly opened window, w/o having to navigate or open a new tab there.

In step 5, I expect existing new windows to apply the icon specified in subsequent `setIcon()` calls.

Tested with Safari 26.5 on Tahoe and Sequoia.

Reproduce the issue with an MV2 extension using browserAction.setIcon({ path: "icons/new_icon.svg" }) and compare existing windows with newly opened windows. Check whether the dynamically selected icon appears only after navigation or opening another tab. Done means subsequent setIcon() calls apply to existing new windows without either action, and the reported behavior is verified on the tested Safari versions.

Written by the indexing model from the issue text.