ref(browser): Streamline browser init() checks

[mydea]

Extracted this out of #15307

This PR:

1. Moves things around in the sdk.ts file a bit, moving types & consts to the top of the file
2. Removes an "unecessary" log around fetch not being supported: This log has two problems. First, it is not actually logged, because we do logger.log() before running init(), which would actually set the logger up. So the log would never show up. Second, the log is superfluous, if the (default) fetch transport fails to send because fetch is not available, you get a warning anyhow for this.
3. Streamlines the browser extension check code slightly.

Closes #16284

[github-actions]

size-limit report 📦

Path	Size	% Change	Change
@sentry/browser	23.95 kB	-0.39%	-92 B 🔽
@sentry/browser - with treeshaking flags	23.71 kB	+0.05%	+10 B 🔺
@sentry/browser (incl. Tracing)	38.36 kB	-0.26%	-98 B 🔽
@sentry/browser (incl. Tracing, Replay)	76.47 kB	-0.16%	-117 B 🔽
@sentry/browser (incl. Tracing, Replay) - with treeshaking flags	69.58 kB	-0.04%	-26 B 🔽
@sentry/browser (incl. Tracing, Replay with Canvas)	81.25 kB	-0.13%	-102 B 🔽
@sentry/browser (incl. Tracing, Replay, Feedback)	93.35 kB	-0.12%	-112 B 🔽
@sentry/browser (incl. Feedback)	40.72 kB	-0.28%	-113 B 🔽
@sentry/browser (incl. sendFeedback)	28.67 kB	-0.4%	-115 B 🔽
@sentry/browser (incl. FeedbackAsync)	33.56 kB	-0.3%	-98 B 🔽
@sentry/react	25.73 kB	-0.35%	-88 B 🔽
@sentry/react (incl. Tracing)	40.34 kB	-0.24%	-96 B 🔽
@sentry/vue	28.34 kB	-0.34%	-94 B 🔽
@sentry/vue (incl. Tracing)	40.19 kB	-0.24%	-93 B 🔽
@sentry/svelte	23.97 kB	-0.4%	-95 B 🔽
CDN Bundle	25.23 kB	-0.02%	-5 B 🔽
CDN Bundle (incl. Tracing)	38.52 kB	-0.04%	-13 B 🔽
CDN Bundle (incl. Tracing, Replay)	74.3 kB	-0.02%	-11 B 🔽
CDN Bundle (incl. Tracing, Replay, Feedback)	79.75 kB	-0.02%	-8 B 🔽
CDN Bundle - uncompressed	73.63 kB	-	-
CDN Bundle (incl. Tracing) - uncompressed	114.03 kB	-	-
CDN Bundle (incl. Tracing, Replay) - uncompressed	228 kB	-	-
CDN Bundle (incl. Tracing, Replay, Feedback) - uncompressed	240.82 kB	-	-
@sentry/nextjs (client)	41.99 kB	-0.22%	-92 B 🔽
@sentry/sveltekit (client)	38.84 kB	-0.3%	-113 B 🔽
@sentry/node	149.47 kB	-	-
@sentry/node - without tracing	98.1 kB	+0.01%	+1 B 🔺
@sentry/aws-serverless	123.43 kB	-	-

View base workflow run

[Lms24]

Nice cleanup, thanks!