[Task]: Refactor code sharing frontend

### What needs to happen?

Currently user code is shared in `SnippetSaveAndShareTabs` which is a wrong place for this.

The right way is to track the state of saving in `PlaygroundController` or `SnippetEditingController` and to always have the shareable descriptor there if it was requested by a widget. This way also the requests will not be duplicated if the sharing dropdown is closed and reopened multiple times.

### Issue Priority

Priority: 3 (nice-to-have improvement)

### Issue Components

- [ ] Component: Python SDK
- [ ] Component: Java SDK
- [ ] Component: Go SDK
- [ ] Component: Typescript SDK
- [ ] Component: IO connector
- [ ] Component: Beam examples
- [X] Component: Beam playground
- [ ] Component: Beam katas
- [ ] Component: Website
- [ ] Component: Spark Runner
- [ ] Component: Flink Runner
- [ ] Component: Samza Runner
- [ ] Component: Twister2 Runner
- [ ] Component: Hazelcast Jet Runner
- [ ] Component: Google Cloud Dataflow Runner