fix: make the run stage persistent

[BrickerP]

What changed

• keep a visible Run Stage between the chart and Lifetime Archive
• restore focused deep links directly into a record-ready stage
• reveal exact data-backed run results, replay the stage, or choose another route
• keep a single accessible stage announcement and collapse the archive below it

Validation

• static P0/P1 review: clear
• git diff --check: pass
• frontend tests and production build: GitHub Actions only