docs: README 10/10 redesign

[OneStepAt4time]

Summary

• Add Table of Contents, 5 badges (npm, downloads, CI, TypeScript, license), and demo GIF
• Restructure Quick Start (prerequisites first, merged headings, GitHub Alert tips)
• Convert Research Modes and Session tools to table format
• Add GitHub Alerts, collapsible Configuration/Compatibility sections
• Add tech stack badges to Architecture, expand Contributing, add footer
• Remove obsolete plan files

Test plan

• Verify README renders correctly on GitHub (dark theme)
• Confirm all ToC anchor links work
• Confirm collapsible sections expand/collapse
• Confirm demo GIF plays
• Confirm all badges render