Use playwright instead of puppeteer in `@sentry/wasm` tests

[AbhiPrasad]

Problem Statement

We use playwright for our Remix and browser integration tests (and we should also do it for our NextJS ones as well), but we don't use it for our @sentry/wasm integration tests.

Solution Brainstorm

Remove puppeteer and use playwright. Removing the dependency reduces dependencies we rely on, which should speed up CI, and make debugging easier across the board.

[smeubank]

would be interested if we have a goal to get rid of pupeteer all together, to have one parent issue listing the remaining using pupeteer to checklist them off

[AbhiPrasad]

With this and #6053 (in progress), we should be getting rid of puppeteer across the board.

Only references to puppeteer in the codebase are in the nextjs and wasm packages.

[github-actions]

This issue has gone three weeks without activity. In another week, I will close it.

But! If you comment or otherwise update it, I will reset the clock, and if you label it Status: Backlog or Status: In Progress, I will leave it alone ... forever!

---

"A weed is but an unloved flower." ― Ella Wheeler Wilcox 🥀