[v8] Remove/update old browser tracing implementations #10898

While we have removed the BrowserTracing exports for all packages now (I believe) in v8, we still have the old implementations lying around, and/or the multi-purpose implementations.

We need to update them to only use/rely on the new implementation, and also get rid of any deprecated stuff in there (e.g. tags and other things that startSpan will not support.

### Tasks
- [ ] https://github.com/getsentry/sentry-javascript/pull/11021
- [x] Refactor usage in Next.js
- [x] Refactor usage in React
- [ ] https://github.com/getsentry/sentry-javascript/pull/10933
- [x] Refactor usage in Sveltekit
- [x] Refactor usage in Vue