ref(browser): Refactor browser integrations to avoid setupOnce

[mydea]

Also a small core refactor...

[AbhiPrasad]

I think we can just remove this tbh, nobody should be relying on it.

[mydea]

yeah, I'll remove it in a next step where I rewrite this to functional style!

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser (incl. Tracing, Replay, Feedback) - Webpack (gzipped)	75.14 KB (-0.03% 🔽)
@sentry/browser (incl. Tracing, Replay) - Webpack (gzipped)	66.57 KB (-0.02% 🔽)
@sentry/browser (incl. Tracing, Replay) - Webpack with treeshaking flags (gzipped)	60.16 KB (-0.03% 🔽)
@sentry/browser (incl. Tracing) - Webpack (gzipped)	31.23 KB (-0.12% 🔽)
@sentry/browser (incl. Feedback) - Webpack (gzipped)	29.85 KB (-0.09% 🔽)
@sentry/browser - Webpack (gzipped)	21.53 KB (-0.14% 🔽)
@sentry/browser (incl. Tracing, Replay, Feedback) - ES6 CDN Bundle (gzipped)	72.57 KB (-0.05% 🔽)
@sentry/browser (incl. Tracing, Replay) - ES6 CDN Bundle (gzipped)	64.28 KB (-0.05% 🔽)
@sentry/browser (incl. Tracing) - ES6 CDN Bundle (gzipped)	30.54 KB (-0.11% 🔽)
@sentry/browser - ES6 CDN Bundle (gzipped)	22.59 KB (-0.21% 🔽)
@sentry/browser (incl. Tracing, Replay) - ES6 CDN Bundle (minified & uncompressed)	202.16 KB (-0.06% 🔽)
@sentry/browser (incl. Tracing) - ES6 CDN Bundle (minified & uncompressed)	92.26 KB (-0.16% 🔽)
@sentry/browser - ES6 CDN Bundle (minified & uncompressed)	67.31 KB (-0.21% 🔽)
@sentry/browser (incl. Tracing) - ES5 CDN Bundle (gzipped)	33.4 KB (-0.22% 🔽)
@sentry/react (incl. Tracing, Replay) - Webpack (gzipped)	66.92 KB (-0.03% 🔽)
@sentry/react - Webpack (gzipped)	21.57 KB (-0.15% 🔽)
@sentry/nextjs Client (incl. Tracing, Replay) - Webpack (gzipped)	83.71 KB (-0.02% 🔽)
@sentry/nextjs Client - Webpack (gzipped)	48.39 KB (-0.06% 🔽)
@sentry-internal/feedback - Webpack (gzipped)	16.18 KB (0%)