The helpful person at #kernelnewbies on OFTC (irc) was kind enough to give me an answer.
ping sets cap_net_raw in the efficient set, creates a socket, then cap_net_raw falls, as seen with strace:
$ strace -e socket,capset ping -c1 localhost capset({_LINUX_CAPABILITY_VERSION_3, 0}, {CAP_NET_RAW, CAP_NET_ADMIN|CAP_NET_RAW, 0}) = 0 socket(PF_INET, SOCK_RAW, IPPROTO_ICMP) = 3 capset({_LINUX_CAPABILITY_VERSION_3, 0}, {0, CAP_NET_ADMIN|CAP_NET_RAW, 0}) = 0
Once the socket is open, no privileges are required to write it anymore.
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