ref(core): Refactor core integrations to avoid setupOnce

[mydea]

Where possible, we should use the new processEvent and/or setup hooks of the integrations, instead of setupOnce. This updates this for the core integrations.

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser (incl. Tracing, Replay, Feedback) - Webpack (gzipped)	75.04 KB (0%)
@sentry/browser (incl. Tracing, Replay) - Webpack (gzipped)	66.38 KB (0%)
@sentry/browser (incl. Tracing, Replay) - Webpack with treeshaking flags (gzipped)	59.99 KB (0%)
@sentry/browser (incl. Tracing) - Webpack (gzipped)	31.13 KB (0%)
@sentry/browser (incl. Feedback) - Webpack (gzipped)	29.74 KB (0%)
@sentry/browser - Webpack (gzipped)	21.39 KB (0%)
@sentry/browser (incl. Tracing, Replay, Feedback) - ES6 CDN Bundle (gzipped)	71.61 KB (0%)
@sentry/browser (incl. Tracing, Replay) - ES6 CDN Bundle (gzipped)	63.33 KB (0%)
@sentry/browser (incl. Tracing) - ES6 CDN Bundle (gzipped)	29.67 KB (0%)
@sentry/browser - ES6 CDN Bundle (gzipped)	21.72 KB (0%)
@sentry/browser (incl. Tracing, Replay) - ES6 CDN Bundle (minified & uncompressed)	199.58 KB (0%)
@sentry/browser (incl. Tracing) - ES6 CDN Bundle (minified & uncompressed)	89.94 KB (0%)
@sentry/browser - ES6 CDN Bundle (minified & uncompressed)	64.92 KB (0%)
@sentry/browser (incl. Tracing) - ES5 CDN Bundle (gzipped)	32.34 KB (0%)
@sentry/react (incl. Tracing, Replay) - Webpack (gzipped)	66.75 KB (0%)
@sentry/react - Webpack (gzipped)	21.44 KB (0%)
@sentry/nextjs Client (incl. Tracing, Replay) - Webpack (gzipped)	83.48 KB (0%)
@sentry/nextjs Client - Webpack (gzipped)	48.23 KB (0%)
@sentry-internal/feedback - Webpack (gzipped)	16.19 KB (0%)