examples: update gatekeeper auth client to use `onError`. #1440

Job	Run time
List examples	10s
List packages	5s
Check packages/typescript-client	27s
Build and test (packages/typescript-client)	1m 19s
Check packages/react-hooks	34s
Build and test (packages/react-hooks)	1m 14s
Check and build examples/proxy-auth	41s
Check and build examples/redis-sync	31s
Check and build examples/bash-client	25s
Check and build examples/linearlite	48s
Check and build examples/gatekeeper-auth	29s
Check and build examples/todo-app	38s
Check and build examples/tanstack-example	38s
Check and build examples/nextjs-example	1m 1s
Check and build examples/basic-example	42s
Check and build examples/remix-basic	38s
	10m 20s

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