ci: Refactor deprecated functionality

[mydea]

See https://github.blog/changelog/2022-10-11-github-actions-deprecating-save-state-and-set-output-commands/ - this will be (probably) removed in March 2023.

[AbhiPrasad]

TIL!

[mydea]

TIL!

Yeah, just noticed it while trying to debug some CI stuff 😅

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser - ES5 CDN Bundle (gzipped + minified)	19.83 KB (+0.02% 🔺)
@sentry/browser - ES5 CDN Bundle (minified)	61.47 KB (0%)
@sentry/browser - ES6 CDN Bundle (gzipped + minified)	18.5 KB (+0.03% 🔺)
@sentry/browser - ES6 CDN Bundle (minified)	54.77 KB (0%)
@sentry/browser - Webpack (gzipped + minified)	20.21 KB (0%)
@sentry/browser - Webpack (minified)	66.17 KB (0%)
@sentry/react - Webpack (gzipped + minified)	20.24 KB (0%)
@sentry/nextjs Client - Webpack (gzipped + minified)	47.48 KB (0%)
@sentry/browser + @sentry/tracing - ES5 CDN Bundle (gzipped + minified)	26.74 KB (+0.01% 🔺)
@sentry/browser + @sentry/tracing - ES6 CDN Bundle (gzipped + minified)	25.03 KB (+0.02% 🔺)
@sentry/replay ES6 CDN Bundle (gzipped + minified)	43.27 KB (-0.82% 🔽)
@sentry/replay - Webpack (gzipped + minified)	38.04 KB (-1.01% 🔽)
@sentry/browser + @sentry/tracing + @sentry/replay - ES6 CDN Bundle (gzipped + minified)	60.53 KB (-0.58% 🔽)