[db-sync] Roll out higher latency dashboard websocket and `api/*` connections to the webapp team

[andrew-farries]

[andrew-farries]

Putting this back into scheduled as we're blocked on the slow workspace starts. We can't roll out the experiment to the team while it has such a dramatic effect on workspace start times.

Start time issue: #15027

[andrew-farries]

Back to in progress now that #15106 should resolve the workspace start issues.