I solved my issue! I isolated it to the set_fixed_mac call in the armbian-firstrun service, specifically after the "nmcli connection down <uuid>" call.
I use ip=dhcp in my kernel command line and at this point the root filesystem is mounted over the active ethernet connection. I don't really understand the purpose of set_fixed_mac, but bringing the connection down is not going to work with nfsroot. I've commented out that set_fixed_mac call and now everything seems to work. Happy days!