docs(js): update Browser JS quick start guide

[inventarSarah]

DESCRIBE YOUR PR

This branch contains the updated Quick Start guide for Browser JS.

The guide uses the platforms/javascript/common/index.mdx file, which contains many includes for the different frameworks. I think the frameworks no longer use this file because each has its own now. I'm not sure if this is used elsewhere, though, so I didn't touch the file much and moved the quick start guide into an include.

@chargome, do you know if we can clean up this file and remove the framework includes and related stuff?

Closes: #15331

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Project	Deployment	Preview	Comments	Updated (UTC)
sentry-docs	Ready	Preview	Comment	Oct 30, 2025 9:58am

1 Skipped Deployment

Project	Deployment	Preview	Comments	Updated (UTC)
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[codecov]

Bundle Report

Changes will increase total bundle size by 201 bytes (0.0%) ⬆️. This is within the configured threshold ✅

Detailed changes

Bundle name	Size	Change
sentry-docs-client-array-push	10.16MB	-6 bytes (-0.0%) ⬇️
sentry-docs-server-cjs	13.03MB	207 bytes (0.0%) ⬆️

Affected Assets, Files, and Routes:

view changes for bundle: sentry-docs-server-cjs

Assets Changed:

Asset Name	Size Change	Total Size	Change (%)
1729.js	-3 bytes	1.78MB	-0.0%
../instrumentation.js	-3 bytes	1.1MB	-0.0%
9523.js	-3 bytes	1.08MB	-0.0%
../app/[[...path]]/page.js.nft.json	72 bytes	877.16kB	0.01%
../app/platform-redirect/page.js.nft.json	72 bytes	877.08kB	0.01%
../app/sitemap.xml/route.js.nft.json	72 bytes	874.3kB	0.01%

view changes for bundle: sentry-docs-client-array-push

Assets Changed:

Asset Name	Size Change	Total Size	Change (%)
static/chunks/pages/_app-*.js	-3 bytes	882.71kB	-0.0%
static/chunks/8321-*.js	-3 bytes	425.87kB	-0.0%
server/middleware-*.js	6.46kB	7.46kB	645.5% ⚠️
server/middleware-*.js	-6.46kB	1.0kB	-86.59%
static/MAES-*.js (New)	77 bytes	77 bytes	100.0% 🚀
static/MAES-*.js (New)	684 bytes	684 bytes	100.0% 🚀
static/x1jqul0SZumtD3sP4RvL0/_buildManifest.js (Deleted)	-684 bytes	0 bytes	-100.0% 🗑️
static/x1jqul0SZumtD3sP4RvL0/_ssgManifest.js (Deleted)	-77 bytes	0 bytes	-100.0% 🗑️