ref(nextjs): Remove internally used deprecated APIs

[lforst]

This PR gets rid of a bunch of internally used deprecated APIs in the serverside of the Next.js SDK. Functionally or API-wise, nothing should change!

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser (incl. Tracing, Replay, Feedback) - Webpack (gzipped)	78.33 KB (+0.17% 🔺)
@sentry/browser (incl. Tracing, Replay) - Webpack (gzipped)	69.57 KB (+0.2% 🔺)
@sentry/browser (incl. Tracing, Replay with Canvas) - Webpack (gzipped)	73.5 KB (+0.19% 🔺)
@sentry/browser (incl. Tracing, Replay) - Webpack with treeshaking flags (gzipped)	63.18 KB (+0.22% 🔺)
@sentry/browser (incl. Tracing) - Webpack (gzipped)	33.52 KB (+0.41% 🔺)
@sentry/browser (incl. browserTracingIntegration) - Webpack (gzipped)	33.39 KB (+0.41% 🔺)
@sentry/browser (incl. Feedback) - Webpack (gzipped)	31.37 KB (+0.11% 🔺)
@sentry/browser (incl. sendFeedback) - Webpack (gzipped)	31.37 KB (+0.1% 🔺)
@sentry/browser - Webpack (gzipped)	22.64 KB (+0.17% 🔺)
@sentry/browser (incl. Tracing, Replay, Feedback) - ES6 CDN Bundle (gzipped)	76.36 KB (+0.25% 🔺)
@sentry/browser (incl. Tracing, Replay) - ES6 CDN Bundle (gzipped)	67.89 KB (+0.25% 🔺)
@sentry/browser (incl. Tracing) - ES6 CDN Bundle (gzipped)	33.66 KB (+0.5% 🔺)
@sentry/browser - ES6 CDN Bundle (gzipped)	24.73 KB (+0.3% 🔺)
@sentry/browser (incl. Tracing, Replay) - ES6 CDN Bundle (minified & uncompressed)	214.06 KB (+0.18% 🔺)
@sentry/browser (incl. Tracing) - ES6 CDN Bundle (minified & uncompressed)	101.87 KB (+0.38% 🔺)
@sentry/browser - ES6 CDN Bundle (minified & uncompressed)	74.25 KB (+0.34% 🔺)
@sentry/browser (incl. Tracing) - ES5 CDN Bundle (gzipped)	36.8 KB (+0.47% 🔺)
@sentry/react (incl. Tracing, Replay) - Webpack (gzipped)	69.93 KB (+0.2% 🔺)
@sentry/react - Webpack (gzipped)	22.67 KB (+0.17% 🔺)
@sentry/nextjs Client (incl. Tracing, Replay) - Webpack (gzipped)	87.55 KB (+0.15% 🔺)
@sentry/nextjs Client - Webpack (gzipped)	51.72 KB (+0.3% 🔺)
@sentry-internal/feedback - Webpack (gzipped)	17.21 KB (0%)