Qiskit / qiskit-ibm-runtime / 36589554939

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Avoid BaseBackendClient class (#3396)

### Summary

The `BaseBackendClient` has only one implementation, and it deviates
from the original spirit (`RuntimeClient` extends the scope as a client
not only for backend-related functionality, and it already diverges the
signature of the base class in one of the methods since `calibration_id`
was introduced).

This PR removes the class, flattening the hierarchy in the process as
the package would end up with just one module.

### Details and comments

Related to #3385

### AI/LLM disclosure

- [x] I didn't use LLM tooling, or only used it privately.

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