extensible-nav - Follow GitHub's nav when it updates after load - #10328

Created by Claude, will be reviewed by @fregante first.

Re-reads GitHub's native nav whenever its current tab or its limited-PRs icon changes, so the extensible nav no longer keeps whatever GitHub had rendered at the moment it first read it. DomChef now re-renders when its as changes, because it only rendered once.

Fixes #10122
Fixes #10037

Test URLs

https://github.com/refined-github/refined-github
https://github.com/refined-github/refined-github/pulse
High counters https://github.com/microsoft/vscode
Blocked PRs https://github.com/sindresorhus/eslint-plugin-unicorn

Also the repro from #10037: https://github.com/sindresorhus/awesome

Possibly related, not touched here: #10317

Checked

Chromium, logged out, with the built extension on the server-rendered markup of https://github.com/refined-github/refined-github and GitHub's scripts blocked. I then mutated the native nav the way GitHub would after the extension has read it, on main and on this branch:

• Moving aria-current from Code to Issues: main keeps Code selected, this branch selects Issues.
• Swapping the Pull requests icon for the locked one (markup copied from https://github.com/sindresorhus/awesome): main keeps the normal icon, this branch shows the locked one.

Not checked (reviewer tasks)

• I couldn't reproduce either issue as GitHub triggers them (real navigation between tabs, Safari cold load). The fix is based on the mechanism above, not on seeing the original race happen. The unrelated console error from active tab of the extensible-nav sometimes lags behind #10122 is not addressed.
• Please check logged in, in Safari and Chrome, that clicking between tabs on the Test URLs always highlights the right tab on the first click, and that https://github.com/sindresorhus/awesome shows the locked icon on a cold load.