feat(browserprofiling): Add manual mode and deprecate old profiling

[s1gr1d]

Adds the manual mode for profiling and browser integration tests.

⚠️ It's still a draft because I need to commit the export of the public APIs and still test some things

Part of #17279

[github-actions]

size-limit report 📦

Path	Size	% Change	Change
@sentry/browser	24.6 kB	-	-
@sentry/browser - with treeshaking flags	23.09 kB	-	-
@sentry/browser (incl. Tracing)	41.26 kB	-	-
@sentry/browser (incl. Tracing, Profiling)	45.75 kB	+0.48%	+216 B 🔺
@sentry/browser (incl. Tracing, Replay)	79.73 kB	-	-
@sentry/browser (incl. Tracing, Replay) - with treeshaking flags	69.4 kB	-	-
@sentry/browser (incl. Tracing, Replay with Canvas)	84.42 kB	-	-
@sentry/browser (incl. Tracing, Replay, Feedback)	96.58 kB	-	-
@sentry/browser (incl. Feedback)	41.27 kB	-	-
@sentry/browser (incl. sendFeedback)	29.27 kB	-	-
@sentry/browser (incl. FeedbackAsync)	34.2 kB	-	-
@sentry/react	26.29 kB	-	-
@sentry/react (incl. Tracing)	43.22 kB	-	-
@sentry/vue	29.09 kB	-	-
@sentry/vue (incl. Tracing)	43.03 kB	-	-
@sentry/svelte	24.61 kB	-	-
CDN Bundle	26.9 kB	-	-
CDN Bundle (incl. Tracing)	41.81 kB	-	-
CDN Bundle (incl. Tracing, Replay)	78.33 kB	-	-
CDN Bundle (incl. Tracing, Replay, Feedback)	83.81 kB	-	-
CDN Bundle - uncompressed	78.84 kB	-	-
CDN Bundle (incl. Tracing) - uncompressed	124 kB	-	-
CDN Bundle (incl. Tracing, Replay) - uncompressed	240.03 kB	-	-
CDN Bundle (incl. Tracing, Replay, Feedback) - uncompressed	252.79 kB	-	-
@sentry/nextjs (client)	45.34 kB	-	-
@sentry/sveltekit (client)	41.64 kB	-	-
@sentry/node-core	50.81 kB	-	-
@sentry/node	157.99 kB	-	-
@sentry/node - without tracing	92.69 kB	-	-
@sentry/aws-serverless	106.45 kB	-	-

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