Improve JSON validation in `start-proxy` action #9763

Job	Run time
check-codeql-versions	49s
analyze-actions	1m 1s
analyze-javascript (ubuntu-22.04)	3m 56s
analyze-javascript (ubuntu-22.04, linked)	4m 17s
analyze-javascript (ubuntu-24.04, linked)	3m 59s
analyze-javascript (windows-2022)	5m 57s
analyze-javascript (windows-2025, linked)	7m 59s
analyze-javascript (ubuntu-24.04)	3m 59s
analyze-javascript (macos-14, linked)	5m 14s
analyze-javascript (macos-13)	6m 30s
analyze-javascript (windows-2025)	5m 34s
analyze-javascript (macos-14)	5m 1s
analyze-javascript (macos-13, linked)	7m 16s
analyze-javascript (macos-15)	4m 12s
analyze-javascript (macos-15, linked)	4m 46s
analyze-javascript (windows-2022, linked)	8m 27s
	1h 18m 57s

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