[Bug]: Sidebar V2 header packs Search and project filter too tightly

[levelorbit]

Before submitting

• I searched existing issues and did not find a duplicate.
• I included enough detail to reproduce or investigate the problem.

Area

apps/web

Steps to reproduce

1. Open T3 Code nightly with Sidebar V2 enabled and at least one project added.
2. Look at the top of the sidebar: Search + New thread on the first row, All projects + Add project on the second.
3. Try to quickly tell the rows apart, then hit the right-side icons (New thread vs Add project).

Expected behavior

Those actions should read as two distinct jobs (global search/new thread vs project scope/add project), with enough visual separation and hit spacing that they’re easy to scan and click reliably.

Actual behavior

Sidebar V2 stacks two near-identical rows:

1. Search + New thread
2. All projects dropdown + Add project

Same structure (wide left control, small right icon) with almost no separation. They're hard to tell apart at a glance and easy to mis-click (especially New thread vs Add project). Follow-up to #4300 after chips were replaced by the dropdown.

Impact

Minor bug or occasional failure

Version or commit

0.0.29-nightly.20260723.882 (b44ed83)

Environment

No response

Logs or stack traces

Screenshots, recordings, or supporting files

file-35e26df53ac2bb964b1790259c7bf25b.png

Workaround

No response