[3.4] chore(e2e): backport TestNoErrorLogsDuringNormalOperations test #15541

Job	Run time
test (linux-amd64-fmt)	56s
test (linux-amd64-integration-1-cpu)	9m 48s
test (linux-amd64-integration-2-cpu)	9m 22s
test (linux-amd64-integration-4-cpu)	9m 43s
test (linux-amd64-functional)	8m 1s
test (linux-amd64-unit-4-cpu-race)	54s
test (all-build)	3m 27s
test (linux-amd64-grpcproxy)	15m 36s
test (linux-amd64-e2e)	14m 7s
test (linux-386-unit)	39s
	1h 12m 33s

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