Remove blocking_sync usages from TaskProducerContext #140

Job	Run time
Test / macOS (Xcode ${{ matrix.xcode_version }} - ${{ matrix.os_version }} - ${{ matrix.arch }})	0s
Test / Linux (nightly-main - noble)	12m 27s
Test / Windows (nightly-main - windows-2019)	51m 53s
cmake-smoke-test / macOS (Xcode ${{ matrix.xcode_version }} - ${{ matrix.os_version }} - ${{ matrix.arch }})	0s
Test / Linux (nightly-main - jammy)	12m 48s
Test / Linux (nightly-main - focal)	12m 44s
cmake-smoke-test / Linux (nightly-main - noble)	6m 18s
Test / Linux (nightly-main - rhel-ubi9)	11m 56s
Test / Linux (nightly-6.2 - noble)	12m 25s
Test / Linux (nightly-6.2 - jammy)	12m 29s
Test / Linux (nightly-6.2 - focal)	13m 9s
Test / Linux (nightly-6.2 - rhel-ubi9)	12m 12s
Soundness / API breakage check	0s
Soundness / Documentation check	1m 49s
Soundness / Unacceptable language check	3s
Soundness / Broken symlinks check	3s
Soundness / License headers check	15s
Soundness / Format check	0s
Soundness / Shell check	1m 51s
Soundness / Python lint check	7s
Soundness / YAML lint check	5s
	2h 42m 34s

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