OK now I have a better picture of it :
When the PC boots, until it reaches the syslinux menu (where you choose 15/25/31 etc ...) : that's the good old BIOS resolution (640x480 at minimum) unless you have a AMD GFX card flashed with atom15. So, until you select one of the horizontal frequencies, you're at 31kHz. No difference with Windows here.
Once you've selected which mod to boot, the linux kernel loads and sets the desired resolution. From now on, you're supposed to be on 15kHz.
Now, a little more details, as I've tried with multiple monitors too just now :
* 15kHz means 640x480i. For a reason that I ignore, my 31kHz couldn't display either the resolution (never had that problem before, but I hadn't tried such a setup for a while). And I noticed the kernel trimmed the available resolutions on the 31kHz and just left anything lower than 640x480 ... weird ... Anyway it explains why your LCD doesn't display anything : it can't do 640x480i
* you're not on full analog displays
* 640x480i is perfectly readable on a TV, so you're probably not at that resolution, as anyway the dialog should bigger
* Did you hear GroovyArcade talk to you while detecting screens ? When testing the HDMI, did it say "Found a LCD or modern CRT" ?
* for NVidia GPUs, I strongly recommend you use the NVidia setting, not the 15kHz. The reason is that the Nvidia driver rejects any resolution with a pixel clock lower than 25MHz, and the Nvidia option sets a 1280x480i which is fine regarding the pixel clock requirement.
* you're using a HDMI transcoder which could also have some limitations (HDMI pixel clock is supposed to be > 25MHz)
I think we get a clearer state of the situation : you used the 15kHz setting, the kernel rejected the resolution, and set a nother one and your transcoder made wonders converting the video back to 15kHz. that's the most probable scenario. So what you should do is try the Nvidia option and tell me how it looks like.
Here is an old video (the detection is slightly faster today, but overall it's pretty much the same that happens) I made a while ago where you can see all steps with a perfectly readbale 640x480i on a CRT using the DVI-I output + DVI-I to VGA + VGA2SCART cable
EDIT :
OMG I've just noticed you are not using MY iso but the old one from VeS (which is not maintained at all, just drop it) ... Try with this one :
https://github.com/substring/os/releases/latest