Some fonts, like Apple Color Emoji, just look like Helvetica.

KC410 wrote: For instance if I make Apple Color Emoji the font in a TextEdit document, the font looks very Helvetica like.

Some fonts have characters which are beyond the capability of the standard US input source to produce. Apple Color Emoji contains emoji which you produce via the Character Viewer app. If you try to type Latin characters, some other font will be used. The same thing can happen with fonts designed for, Arabic, Chinese, etc. In order to type what is in such fonts, you may need to use a different input source. Under the Unicode system that computers normally use, you switch input sources to write in symbols and different scripts and the fonts usually take care of themselves.

Use emoji and symbols on Mac – Apple Support (UK)

Write in another language on Mac – Apple Support (UK)

KC410 wrote:

what is typed gets "pulled" from the first 128 unicode characters (or slightly highter if the option key is used), and the more picturesque forms are stored higher up in the list, out of reach of the regular keyboard.

Yes, but to be clear, your hardware keyboard can type any of the 150,000 unicode characters, just by switching the input sources provided by MacOS in system settings > keyboard. If you want you can create a custom input source that types any collection of characters you choose with the Ukelele app.