Realtek r8127 Ethernet NIC disappears on every reboot, only recovers with full power cycle

Could you please check what driver you are using (lspci -vv)?

Kernel driver in use: r8127
Kernel modules: r8169, r8127

It should be r8127. The r8169 driver causes the device to disappear after soft-reboot. Simply blacklist the driver, e.g. add modprobe.blacklist=r8169 to the kernel command line.