How do I enlarge Mathematica's GUI font size?
The text should be using the ambient font and size as spec'd from the OS. What is returned from evaluating CurrentValue["SystemFonts"]? The output from CurrentValue["ConnectedDisplays"] might be useful as well.
CurrentValue["SystemFonts"]
CurrentValue["ConnectedDisplays"]
AHHH wait, I did not see your question, good question that needs to be addressed by Wolfram, that splash screen is not a notebook with an editable stylesheet. This is an accessibility bug I think. Please report to Wolfram
Very few GUI interface elements in Mathematica, but many have this issue. There is always an alternative way to access the same functionality without the GUI. I have disabled the Welcome Screen that you show by default and just use the standard menu operations since I do not like the Welcome Screen font sizes or colors.
You can also tweak your stylesheet if you just want some fonts bigger https://reference.wolfram.com/language/tutorial/WorkingWithStylesheets.html
Preferences - Interface - Global Magnification.
I have mine at 150 %
Edit: Sorry, I thought it was the Notebook text size.