chore: extract telemetry_handle creation into helper function (#10008)
### Description
Small refactor pulling the logic for initializing the `telemetry_handle`
and client into a helper function named `initialize_telemetry_client`.
Previously, this initialization logic required a clarifying comment as
well as nested match statements in the body of `run` which required time
to read and understand.
This PR moves those nested match statements into a helper, reducing the
cognitive load while reading `run`. This helper also centralizes logic
for initialization, and allows us to remove a `mut` from
`telemetry_handle`, since it's no longer being modified outside the
helper.