ref(core): Ensure we use captureException() from @sentry/core where possible

[mydea]

This is a reduced version of #6488.

Basically, it makes sure we use the functional captureException() from core wherever possible internally.
There are two main places where we still use hub.captureException internally:

https://github.com/getsentry/sentry-javascript/blob/fn/import-captureException/packages/node/src/integrations/onuncaughtexception.ts#L127

and

https://github.com/getsentry/sentry-javascript/blob/fn/import-captureException/packages/node/src/integrations/onunhandledrejection.ts#L52

Which are both places that provide a hint with handled: false.

ref #6073

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser - ES5 CDN Bundle (gzipped + minified)	19.78 KB (-0.02% 🔽)
@sentry/browser - ES5 CDN Bundle (minified)	61.27 KB (0%)
@sentry/browser - ES6 CDN Bundle (gzipped + minified)	18.57 KB (+0.02% 🔺)
@sentry/browser - ES6 CDN Bundle (minified)	54.8 KB (0%)
@sentry/browser - Webpack (gzipped + minified)	20.34 KB (0%)
@sentry/browser - Webpack (minified)	66.48 KB (0%)
@sentry/react - Webpack (gzipped + minified)	20.36 KB (0%)
@sentry/nextjs Client - Webpack (gzipped + minified)	47.57 KB (0%)
@sentry/browser + @sentry/tracing - ES5 CDN Bundle (gzipped + minified)	26.75 KB (+0.01% 🔺)
@sentry/browser + @sentry/tracing - ES6 CDN Bundle (gzipped + minified)	25.19 KB (-0.01% 🔽)
@sentry/replay ES6 CDN Bundle (gzipped + minified)	41.87 KB (-0.01% 🔽)
@sentry/replay - Webpack (gzipped + minified)	38.1 KB (0%)

[lforst]

Changes look fine to me! Although I don't yet fully understand where we're going with this.