fix(web): resolve healthz cache warning by using lightweight count query #5501

Job	Run time
build-and-check	2m 30s
test-integration-packages/billing	1m 5s
test-integration-packages/agent-runtime	37s
test-integration-web	42s
test-integration-packages/internal	38s
test-integration-cli	34s
test-.agents	36s
test-packages/internal	47s
test-integration-sdk	41s
test-packages/billing	39s
test-cli	1m 11s
test-sdk	51s
test-packages/agent-runtime	1m 3s
test-common	42s
test-web	45s
test-integration-common	38s
	13m 59s

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