ref: Clean up new integrations API and pave migration path

[lforst]

This PR intends to supersede this PR and our current migration path because it is very convoluted and leads to potential typing issues. #10143

What is different?

• We keep all the integration classes as we had them before (they need to stick around for v7 in any case - we can remove them in v8). This sadly involves reverting all of our class-removal work, however, the work wasn't entirely in-vain, as all the refactoring to reduce state will help us when we completely move the logic from classes to functions.
• We introduce a helper function similar to rollup and vite's defineConfig() function that let's us and users create integration factories in a type-safe manner.
• We expose functional-alternatives for all of our integrations that do nothing else than simply instantiating the class equivalent. For this, we can also use the helper function.

What will this do?

• Will keep the typings for our class-based integrations.
• The return types of the functional integrations will not be widened to Integration but will exactly represent the return value as it is. We can evaluate whether this is beneficial for us because it, in theory and counter-intuitively, widens our API. We could also decide on an integration-by-integration basis whether an integration should be of the type Integration or it's actual shape.

Are there any drawbacks?

No.

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser (incl. Tracing, Replay, Feedback) - Webpack (gzipped)	77.05 KB (+0.07% 🔺)
@sentry/browser (incl. Tracing, Replay) - Webpack (gzipped)	68.41 KB (+0.08% 🔺)
@sentry/browser (incl. Tracing, Replay) - Webpack with treeshaking flags (gzipped)	62.05 KB (+0.06% 🔺)
@sentry/browser (incl. Tracing) - Webpack (gzipped)	32.44 KB (+0.16% 🔺)
@sentry/browser (incl. Feedback) - Webpack (gzipped)	31.03 KB (+0.11% 🔺)
@sentry/browser - Webpack (gzipped)	22.37 KB (+0.19% 🔺)
@sentry/browser (incl. Tracing, Replay, Feedback) - ES6 CDN Bundle (gzipped)	74.69 KB (+0.02% 🔺)
@sentry/browser (incl. Tracing, Replay) - ES6 CDN Bundle (gzipped)	66.34 KB (+0.02% 🔺)
@sentry/browser (incl. Tracing) - ES6 CDN Bundle (gzipped)	32.17 KB (+0.07% 🔺)
@sentry/browser - ES6 CDN Bundle (gzipped)	23.98 KB (+0.09% 🔺)
@sentry/browser (incl. Tracing, Replay) - ES6 CDN Bundle (minified & uncompressed)	209.07 KB (+0.07% 🔺)
@sentry/browser (incl. Tracing) - ES6 CDN Bundle (minified & uncompressed)	97.12 KB (+0.15% 🔺)
@sentry/browser - ES6 CDN Bundle (minified & uncompressed)	71.73 KB (+0.2% 🔺)
@sentry/browser (incl. Tracing) - ES5 CDN Bundle (gzipped)	35.18 KB (+0.04% 🔺)
@sentry/react (incl. Tracing, Replay) - Webpack (gzipped)	68.78 KB (+0.06% 🔺)
@sentry/react - Webpack (gzipped)	22.41 KB (+0.18% 🔺)
@sentry/nextjs Client (incl. Tracing, Replay) - Webpack (gzipped)	85.44 KB (+0.05% 🔺)
@sentry/nextjs Client - Webpack (gzipped)	49.55 KB (+0.09% 🔺)
@sentry-internal/feedback - Webpack (gzipped)	17 KB (0%)