CM3 no boot at all (no white screen)

[Flavor]

dd, Etcher, or even the Raspberry Pi Imager should all work. They do the same thing. They take the .img file and 'dd' it to the SD card. You do not need to have any partitions or filesystems on the SD card. Even if you do, they will get overwritten by the dd.

If you use dd, you need to dd to the SD card and not a partition on the card. It would be like
dd if=file.img of=/dev/sdc bs=4M
or some such (written from memory)

You would _NOT_ want to use of=/dev/sdc1 or any number following the device's letter.

Notice, also, that I used the .img file for dd. Etcher and the Raspberry Pi Imager should take the .img.gz and ungzip it to .img during the process. dd is pretty dumb, so it will allow you to give it a .img.gz file, but that won't work well in the end.

gunzip file.img.gz
then use dd as shown above

After you image 'to' the SD card using your Linux machine, what does the SD card look like? Once dd is complete, pop the SD out and then back in. What does 'lsblk' say? Does it show any partitions on the SD card?

How long did you hold down the power button? Do you ever see anything on the HDMI while you're holding the power button in the ON position (for like 10 seconds or so)?

Another thing to try is disconnecting the Raspberry Pi from the Freeplay board. Then, re-connect it. I have heard of rare times when this fixed odd problems.