Out of curiosity, could you try booting another computer off that flash drive - assuming you have another flash-drive friendly computer at hand?

A couple of other things which I've encountered on my own machines or in seeking answers:

a) Might it be worth re-checking your BIOS to see that booting from a USB flash-drive is enabled? (I gather you get into BIOS by tapping DEL during start - no idea whether you need to specify USB flash, floppy, or hard drive, though! - or just something like "USB"))

Alternatively, tap F11 lightly during start - I see that allows you to choose boot options - could just help if you try that with the flash drive plugged in.

b) For some reason, your computer might be struggling to recognise your flash drive as a bootable drive. It could simply be a dodgy flash drive - have you another you could try?

(I'd be inclined to use cheap low capacity drives for initial experiments - 1gb will do for most Unetbootin installs; 4gb should cover just about anything.)

c) I've no idea why, but I've had problems with Unetbootin and flash drives formatted with Linux-style extended file systems - seem to remember NTFS also flopped. It might be worth checking that your flash drive is formatted to FAT32. It almost certainly will be, if it's a smaller drive - but it occurs to me that your 32gb drive could just have been formatted to NTFS. If the flash drive you're using has any other file system, then try re-formatting it to FAT32, then try Unetbootin again.

It really seems strange that something as modern as your Advent should be so otherwise - which makes me hopeful that we're missing some simple, basic point! Might be worth trying the suggestions above (I'm sure you've already tried the BIOS stuff, but the different flash drive/FAT32 format for the flash drive could just help) before seeking more complex solutions.