test: harden three flaky test areas against CI races #9966

Job	Run time
Build (linux/arm64)	13m 25s
Build public push gateway (linux/amd64)	1m 18s
Build public push gateway (linux/arm64)	8m 22s
Build (linux/amd64)	15m 7s
Qualify relay image source	-1s
Publish public push gateway image	0s
Merge ${{ matrix.variant }} multi-arch manifest	0s
	38m 11s

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