Switch CI from pip install to uv sync

[gkorland]

Parent issue

Part of #550 (CI Pipeline Optimization Analysis)

Problem

Both nextjs.yml (Build) and playwright.yml (Playwright Tests) use pip install -e . / pip install -e ".[test]" for backend dependency installation, which takes 27-35s per job.

Solution

Replace pip install with uv sync which is 3-5x faster:

- name: Install uv
  uses: astral-sh/setup-uv@v5
  with:
    version: "latest"

- name: Install dependencies
  run: uv sync

Apply to both nextjs.yml and playwright.yml.

Estimated savings

~15-20s per shard, ~40s total

Effort

Low