I read the question as "what if the designer uses designer tools to make a PNG with text instead of HTML that calls for a fancy font, then posts that?" It seems like that should (in the US) work like any normal use of a font on a designer's pc^Wmac, and the image doesn't get copyright-like encumbrances from the font.

Exactly. I should have been more clear.

This is the right principle to have absorbed from the article linked, but applied incorrectly.

For either of the hypotheticals here, it would be possible to operate in the way described, subject to some constraints. There is no reason to think that the courts would perceive any legal imperative to recognize the distinction between server licenses or otherwise on copyright grounds. Companies can't invent law and say that it applies just because they wrote down how they'd like for things to work.

The technical reasons referenced earlier are what make these hypotheticals untenable, not law.