ref(replay): Extract integration to clarify public API

[mydea]

This moves the actual replay integration code into a dedicated module, making the public API (=what is exported from index.ts) very explicit. Basically, only what is exported from there is considered public.

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser - ES5 CDN Bundle (gzipped + minified)	19.67 KB (-0.01% 🔽)
@sentry/browser - ES5 CDN Bundle (minified)	60.94 KB (0%)
@sentry/browser - ES6 CDN Bundle (gzipped + minified)	18.46 KB (0%)
@sentry/browser - ES6 CDN Bundle (minified)	54.49 KB (0%)
@sentry/browser - Webpack (gzipped + minified)	20.26 KB (0%)
@sentry/browser - Webpack (minified)	66.24 KB (0%)
@sentry/react - Webpack (gzipped + minified)	20.29 KB (0%)
@sentry/nextjs Client - Webpack (gzipped + minified)	47.14 KB (0%)
@sentry/browser + @sentry/tracing - ES5 CDN Bundle (gzipped + minified)	26.52 KB (0%)
@sentry/browser + @sentry/tracing - ES6 CDN Bundle (gzipped + minified)	24.96 KB (+0.02% 🔺)
@sentry/replay ES6 CDN Bundle (gzipped + minified)	41.79 KB (-0.01% 🔽)
@sentry/replay - Webpack (gzipped + minified)	37.91 KB (0%)