Clean up concurrency in NIOEchoClient #1301

Job	Run time
Construct integration test matrix	3s
Vsock tests	2m 11s
Cxx interop / Construct Cxx interop matrix	3s
Benchmarks / Construct Benchmarks matrix	5s
Unit tests / Construct unit test matrix	5s
Soundness / API breakage check	2m 2s
Soundness / Documentation check	2m 6s
Soundness / Unacceptable language check	2s
Soundness / Broken symlinks check	3s
Soundness / License headers check	9s
Soundness / Format check	45s
Soundness / Python lint check	5s
Soundness / Shell check	40s
Soundness / YAML lint check	3s
Integration tests / Linux (5.9)	3m 38s
Integration tests / Linux (5.10)	3m 38s
Integration tests / Linux (6.0)	3m 39s
Integration tests / Linux (nightly-6.1)	3m 37s
Integration tests / Linux (nightly-main)	3m 39s
Cxx interop / Cxx interop / Linux (5.9)	1m 7s
Cxx interop / Cxx interop / Linux (5.10)	1m 4s
Cxx interop / Cxx interop / Linux (6.0)	1m 15s
Cxx interop / Cxx interop / Linux (nightly-6.1)	1m 37s
Cxx interop / Cxx interop / Linux (nightly-main)	1m 29s
Benchmarks / Benchmarks / Linux (5.9)	3m 37s
Benchmarks / Benchmarks / Linux (5.10)	3m 37s
Benchmarks / Benchmarks / Linux (6.0)	3m 37s
Benchmarks / Benchmarks / Linux (nightly-6.1)	3m 37s
Benchmarks / Benchmarks / Linux (nightly-main)	3m 38s
Unit tests / Unit tests / Linux (5.9)	3m 34s
Unit tests / Unit tests / Linux (5.10)	3m 34s
Unit tests / Unit tests / Linux (6.0)	3m 35s
Unit tests / Unit tests / Linux (nightly-6.1)	3m 33s
Unit tests / Unit tests / Linux (nightly-main)	3m 34s
	1h 9m 1s

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