does fly.io have any writable directories on the filesystem?

It does. You can SSH and test writing a file(s):

flyctl ssh console --app <appname> --select

# echo “I am writing to a file” > /dev/shm/file1
# cat /dev/shm/file1
I am writing to a file
# echo “I am writing to a file” > /root/file2
# cat /root/file2
I am writing to a file
# exit

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