How about using the Saved tab as your default in the New Document dialog? You can go through the other tabs and select presets that are useful to you, and save them (you need a different name).
Then select the Saved tab. Unfortunately, after restarting Photoshop and clicking on New opens the dialog in the Recent tab again, so not as tidy a solution as you are probably looking for. As you can see from the dustbin icon in the top right corner of the A4 (custom) preset below, you can delete templates from this list.
You can actually open and edit the Recent Files file in Notepad, but it contains precious little in plain English text.
You have some excellent ideas for how to work around this issue and I am most appreciative.
It is still not possible, even with the Saved tab, to delete any templates downloaded from Adobe, only one's own templates.
I suppose I could delete the Start.psw file and get a new Start page, but I am loathe to touch system files without knowing more about what that might do or not do.
Thank you for your very helpful and very clear response. The screenshots were very useful!
Shame on you Adobe. I'm a software engineer. This was a design choice by them to stop you doing so. Gives you a reason to "upgrade". Good business tho.
To delete templates, on Illustrator CC 2017 on a Windows 10 x64 machine. you need to locate the file:
"C:\Users\tshaw\AppData\Roaming\Adobe\Adobe Illustrator 21 Settings\en_GB\x64\Adobe Illustrator Prefs"
Anyway, once open, do a search for "/artboard" You might need to search a few times, maybe 6-10 times to find what you're looking for.
You might see something like:
Delete the sizes you want and make sure the numbering (height_3 etc, changes those numbers!!) is still correct. This might leave some settings from the presets left in the file but it works...I wonder if Adobe checks their configs for redundant config options...probably not.
Anyway, hope it works for you.
Thank you very much for your helpful comment.
On Mac OSX 10.12.4 I believe the same information is to be foun at /Users/linda/Library/Preferences/Adobe\ Photoshop\ CC\ 2017\ Settings/New\ Doc\ Sizes.json