Design: a new forge's claim link and the client list use the design system, and claim-link-box/forges-sent leave §5d #3871

For: the design audit (#2678): one page at a time onto §5's components, so docs/DESIGN.md §5d only shrinks. A forge in a link is what the owner hands a client; the link and the list of what was sent should look finished.

Done:

• [ ] The claim link on /forges/new's answer and the sent list on /forges is drawn from §5's components (src/web/ui.ts helpers such as row, card, badge, btn, and §5 classes) in place of .claim-link-box .forges-sent, which src/web/style/content.ts defines today. The claim link stays one tap to copy.
• [ ] 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/pages/new-forge.ts, src/web/pages/forges.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 stayed red through 2 automatic tries (pull #3901 closed). Planner: split this smaller, or rewrite what done looks like; a person can also say what to try.

Split into #4172 (the markup) and #4173 (the stylesheet and docs), so the design check stays green between them.