As noted elsewhere my experience of the 5070 is that it will only boot in UEFI mode, not Legacy mode. I followed a similar path to what I did with the Wyse 3040 and checked out Puppy Linux and the 32-bit and 64-bit versions of Tiny Core 12.0 all of which ran without any problems.
The onboard flash appears as /dev/mmcblk0.
UEFI booting describes how I installed Tiny Core to the internal eMMC flash of the 5070.
In October 2025 I heard from Jon:
I recently got a Dell Wyse 5070 (4GB RAM, 32GB emmc) for use as my headless media server. I installed Debian Trixie using the latest Debian netinst iso, burnt to a USB with Balena, and it installed directly to the emmc with no issues at all, just the same as installing on a 'normal' sata disk.