Hi Mike,
So I downloaded and unpacked your portable. But I'm running Bionicpup64 which, according to pupsys-info, is using " Glibc: 2.27". I figure "What the hell.
Right-Clicking 'LAUNCH' and selecting 'Run in Terminal' will only take a moment".
The terminal response was:
"[23910
0319/191931.895802:ERROR:message_box_dialog.cc(143)] Unable to show a dialog outside the UI thread message loop: Failed To Create Data Directory - Chromium cannot read and write to its data directory:
/initrd/mnt/dev_save/Pup-Apps/Ungoogled_Chromium-portable/chromium/PROFILE
[23985
0319/191933.468295:ERROR:sandbox_linux.cc(374)] InitializeSandbox() called with multiple threads in process gpu-process.
[23910
0319/192000.783942:ERROR:ev_root_ca_metadata.cc(841)] Failed to register OID: 0"
But the UG-C opened and I'm posting from it now.
What all the above complaints means in practical terms and how much older the glibc can be are yet to be determined.
But, thanks, Mike.
And pass my appreciation on to Marmaduke if you can.
p.s. Just noticed the curious 'boxes' in the quoted terminal output which, AFAICT, were added by the Forum's software. 
p.p.s. I closed UG-C via Menu>Exit as I wanted to see what the terminal had to say about closing. I didn't have anything unusual to say. Looked in UG-C's "chromium" folder. There's a big red=broken symbolic link to /opt/Ungoogled_Chromium-portable/chromium/PROFILE which makes sense as the portable isn't in /opt. But it shouldn't be there at all. 
p.p.p.s. Simple fix. Deleted the link. Restarted UG-C and a proper PROFILE folder was created.