Make cache clearing robust to directories without read permissions #6156

Job	Run time
cargo fmt	21s
cargo clippy \| ubuntu	52s
cargo clippy \| windows	2m 29s
build binary \| linux	1m 54s
cargo test \| windows	7m 22s
cargo test \| ubuntu	3m 58s
Determine changes	5s
cargo test \| macos	4m 17s
check windows trampoline	4m 29s
build binary \| macos aarch64	1m 13s
build binary \| macos x86_64	1m 51s
build binary \| windows	3m 4s
benchmarks	1m 57s
check cache \| macos aarch64	11s
check system \| homebrew python on macos aarch64	19s
check system \| python on macos aarch64	15s
check system \| python on macos x86_64	24s
check system \| python on debian	48s
check cache \| ubuntu	54s
check system \| python on fedora	32s
ecosystem test \| prefecthq/prefect	2m 5s
check system \| python on opensuse	46s
check system \| pypy on ubuntu	24s
ecosystem test \| pallets/flask	11s
check system \| python on rocky linux 8	40s
check system \| python on ubuntu	15s
check system \| amazonlinux	32s
check system \| alpine	27s
check system \| python on rocky linux 9	34s
check system \| python via pyenv	1m 48s
check system \| pyston	3m 58s
check system \| python3.10 on windows	34s
check system \| python3.13 on windows	1m 26s
check system \| python3.12 via chocolatey	1m 25s
check system \| conda3.8 on linux	39s
check system \| conda3.11 on linux	52s
check system \| embedded python3.10 on windows	25s
check system \| python3.10 on windows x86	1m 1s
check system \| conda3.8 on windows	2m 7s
check system \| conda3.11 on windows	2m 14s
check system \| conda3.8 on macos	43s
check system \| conda3.11 on macos	48s
	1h 1m 9s

Provide feedback