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GH-36369: [C++][FlightRPC] Fix a hang bug in FlightClient::Authenticate*() (#36372)

### Rationale for this change

We need to drain all responses from server before we call gRPC's Finish() to avoid hanging.

### What changes are included in this PR?

Read all responses before calling Finish().

### Are these changes tested?

Yes.

### Are there any user-facing changes?

Yes.
* Closes: #36369

Authored-by: Sutou Kouhei <kou@clear-code.com>
Signed-off-by: Sutou Kouhei <kou@clear-code.com>