Design: a run's check steps and progress use the design system, and check-steps/check-progress leave §5d #3873

For: the design audit (#2678): one page at a time onto §5's components, so docs/DESIGN.md §5d only shrinks. A check's verdict is what lands a pull; its steps should be calm §5 rows, red only where a step failed.

Done:

• [ ] A run's check steps and progress bar is drawn from §5's components (src/web/ui.ts helpers such as row, card, badge, btn, and §5 classes) in place of .check-steps .check-progress, which src/web/style/content.ts defines today. src/web/client.ts selects some of these names: move its selectors to data-* attributes in the same pull, so the live behaviour is kept.
• [ ] Those classes' rules are gone, and their names leave their bullet of docs/DESIGN.md §5d. bun scripts/design-lint.ts passes, and so does --update with a lower or equal baseline.
• [ ] The pull carries before/after screenshots in light and dark, at desktop and 390 px wide. Nothing a person reads or taps is lost, and every form posts the same fields.
• [ ] The page's existing tests pass. A test asserts the old class names are absent from the page's HTML.

Files: src/web/studio/run.ts, src/web/client.ts, src/web/style/content.ts, docs/DESIGN.md, the page's test. Risk: another page borrowing one of these classes. git grep each name first, and if one is used elsewhere, keep it and say so in the pull.

Parked by the crew: the check failed in the place it ran, not in the change, and failed the same way when it was run again (pull #3886 closed). Planner: split this smaller, or rewrite what done looks like; a person can also say what to try.

Steward (trevato): pull #3886 was parked by an infrastructure fault, not this work - the egress cap from #3719 stopped every check's package install (02:20-03:12Z). The forge is rolled back and the cap is fixed (c7ea1935). Reopened and re-checked; this strike does not count.