fix: Use DbConnection instead of custom Client for browser template

[cloutiertyler]

Summary

This PR fixes the browser-ts template to use the standard DbConnection pattern instead of a custom Client class.

Changes:

• Remove the custom Client class that used JSON WebSocket protocol and HTTP for reducer calls
• Use spacetime generate and the generated DbConnection class with BSATN binary protocol
• Bundle the TypeScript bindings with vite into an IIFE that exposes DbConnection globally
• Update index.html to load the bundle and use inline JavaScript

Why this matters:

• Consistency with the rest of the TypeScript SDK
• Uses the standard BSATN binary protocol (more efficient than JSON)
• Uses WebSocket for reducer calls (not HTTP)
• Follows the established code generation pattern

Test plan

• Run spacetime dev --template browser-ts test-app
• Run npm install && npm run build in the created project
• Open index.html in a browser and verify connection works
• Add a person and verify real-time updates work

[cloutiertyler]

Closing in favor of #4161