ref(nextjs): Clean up client-side integrations code

[lobsterkatie]

This refactors the code in the nextjs SDK adding client-side integrations to:

• Consolidate the logic, which currently is split between init and a helper function, to all live in the helper. Not only is this better practice in general, it also sets up for a future PR which will add a new integration. (Adding more code to the current somewhat-messy setup would just make it worse, and it's easier to fix it before that happens.)
• Align the client-side code with the server-side code, for greater maintainability.

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser - ES5 CDN Bundle (gzipped + minified)	19.57 KB (-0.01% 🔽)
@sentry/browser - ES5 CDN Bundle (minified)	60.59 KB (0%)
@sentry/browser - ES6 CDN Bundle (gzipped + minified)	18.36 KB (+0.02% 🔺)
@sentry/browser - ES6 CDN Bundle (minified)	54.15 KB (0%)
@sentry/browser - Webpack (gzipped + minified)	20.12 KB (0%)
@sentry/browser - Webpack (minified)	65.75 KB (0%)
@sentry/react - Webpack (gzipped + minified)	20.14 KB (0%)
@sentry/nextjs Client - Webpack (gzipped + minified)	45.95 KB (-0.04% 🔽)
@sentry/browser + @sentry/tracing - ES5 CDN Bundle (gzipped + minified)	26.42 KB (+0.01% 🔺)
@sentry/browser + @sentry/tracing - ES6 CDN Bundle (gzipped + minified)	24.85 KB (+0.02% 🔺)
@sentry/replay index.js	37.08 KB (-0.33% 🔽)
@sentry/replay - Webpack (gzipped + minified)	37.06 KB (-0.32% 🔽)
@sentry/replay - Webpack (minified)	124.66 KB (-0.27% 🔽)