[diag] Move `otPlatDiagModeGet()` check to `ProcessCmd` #3

Job	Run time
channel-manager-csl	3m 2s
expects	14m 40s
packet-verification-1-1-on-1-4	13m 10s
packet-verification-low-power	9m 57s
thread-1-4-gcc-m32	11m 38s
thread-1-4-posix	10m 10s
thread-1-4-gcc-m64	11m 14s
thread-1-4-clang-m32	9m 47s
thread-1-4-clang-m64	9m 48s
upload-coverage	0s
	1h 33m 26s

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