I have a Netgear WGR614v10. Around August 2016, I tried to replace the factory firmware with lede-brcm47xx-mips74k-netgear-wgr614-v10_north-america-squashfs.chk. This failed.

When the device boots it sends TFTP requests for vmlinuz. Could this allow restoring a firmware? Where can I find a suitable vmlinuz file?

You can try with official recovery article https://kb.netgear.com/22688/How-to-upload-firmware-to-a-NETGEAR-router-using-TFTP

There are some people on dd-wrt forums that have dumped the firmware with tftp AND serial access to the board (this allows you to tell tftp to load whatever you want). http://www.dd-wrt.com/phpBB2/viewtopic.php?p=409803#409803
but there is no info on what are the commands to flash it after you transferred it, nor what is the pinout of the serial. It should be 4 pins as this is TTL serial, not RS232 serial.

By looking at the images here http://infodepot.wikia.com/wiki/Netgear_WGR614_v10 it seems the soldered pins are the TTL but you'd have to use a multimeter to see what is power pins so you can ingore them while you try connecting the TTL-usb dongle to the others until you get the connections right.

From what I read from both the dd-wrt forum and in hardware specs, it's probably not really worth it. 4MiB of flash (no customization) and older broadcom wifi (crappy or non-existent opensource drivers).