Your comments

When I create a new file with Textastic, the emojis work just fine, and they also display correctly on the desktop in Chocolat. Maybe there is something wrong with the editor on the Mac, but since they display fine in Safari, I am not quite sure.
Interestingly, when I open the document on the iPad, remove the "desktop" emojis and insert new ones on the device it shows the same blocks.
Well, I have to find them on the iPad. I thought they had to be the same, otherwise iMessage etc. would be broken too. Will try with a simple smiley. You can add them from the Mac by pressing CMD+ALT+T.

Where did you enter the emojis?

I entered them on the Mac with Chocolat. In there and in the browser they display just fine.
Yeah, I have seen the "show tabs" feature, though I don't use tabs myself.

Anyhow, let's see if someone else finds this useful too.
Editor + Browser split-view could be interesting.