I love this editor, and I'd like to ask for a tiny feature: CMD-V for pasting text.

Cmd-V is definitely there, I use it all the time. Have a look at the menu Edit -> Paste.

I have 4.0.3 (38) on Mojave and Catalina, and I don't see V in the Edit menu, nor does CMD-V do anything with text in the clipboard.

Here's a screenshot of how the menu looks for me on macOS 10.15.4 and Textastic 4.0.3.

I don't see a preference which would be relevant... FWIW, ever since I installed it CMD-V was missing. I ignored the fact for quite some time but thought I should report it now. 

Do you have a third party app installed that could possibly mess with menu items?

None that I know of. Yours is actually one of relatively few third-party apps I use. In particular I've not yet encountered an application into which I can paste text which didn't have support for CMD-V. Assuming I had an app which messed with menu items, I would expect it to mess with them in all apps. :-)

Here's a screenshot of how the menu looks for me on macOS 10.14.6

Ok, so the menu item is there, but the keyboard shortcut is missing.

Please have a look at System Preferences -> Keyboard -> Shortcuts -> App Shortcuts. 

Maybe Cmd-V is set for another menu item or it is disabled for Paste?

This is what I see there

See the problem? Cmd-V is set to "Paste and Match Style" instead of Paste. Since Textastic doesn't have a "Paste and Match Style" menu item, the shortcut for "Paste" is disabled by the system.

I did not see the problem, but thanks to your explanation I now do. (I'd love to know now what program on two distinct Macs did that ..)

I have changed the entry from "Paste and Match Style" to simply "Paste", and I now see apple-V and CMD-V works as I desire.

Thank you very much for your help, Alexander. 

I'm glad that it works now.

Actually, you can just remove that entry in the list of app shortcuts completely if you don't need it.