[v8] Replace custom BrowserTracing routing instrumentations #10387
Description
In v7, new BrowserTracing() accepts a custom routingInstrumentation. We use this in a few places.
The replacement browserTracingIntegration() does not accept this anymore (as it is tightly tied to transactions). Instead, this needs to be refactored to a new format using utility methods to start pageload/transaction spans.
### Tasks
- [ ] https://github.com/getsentry/sentry-javascript/pull/10373
- [ ] https://github.com/getsentry/sentry-javascript/pull/10353
- [ ] https://github.com/getsentry/sentry-javascript/pull/10430
- [x] React Router v3
- [x] React Router v6
- [ ] https://github.com/getsentry/sentry-javascript/pull/10397
- [ ] https://github.com/getsentry/sentry-javascript/pull/10450
- [x] Vue Router Instrumentation
- [ ] https://github.com/getsentry/sentry-javascript/pull/10442