Subtracting two areas is like adding two areas where one is negative…right?

Yes , but see my second paragraph. Just because something is true it doesn't mean it's intuitive.

That's interesting. To me it seems intuitive. It's a real area that can be drawn like any other. The sign is an operator describing what function to visualize, not a property of the measured area. So thinking of it in that way eliminates any need for the term "negative area."

But, intuition is subjective, so you may need to adjust the terminology to fit the visualization.

It is intuitive on some higher level, now that I know about signs, operators, negative numbers and all that. But when talking about visual information (i.e. "visual proof"), it isn't intuitive in that context.

In addition to that, for all I know there could be some pitfalls involved with negative areas which I'm not aware of. Even if there aren't any pitfalls, this isn't immediately obvious to someone who isn't familiar with the concept of negative area.

If I'm willing (or forced) to think in such abstract terms, I would much prefer an algebraic proof to this visual proof.

Your intuition is irrelevant if they’re mathematically equivalent.

There are no serious pitfalls with oriented areas. Adding them to your arsenal of geometric proof tools will greatly simplify many proofs. Not having such a concept makes ancient geometry books much more complicated than they need to be, often requiring lots of detailed case analysis where the separate cases are essentially the same, just oriented opposite ways.

It’s intuitive, just in more dimensions. People have different ideas and abilities to imagine/think dimensions, but on top of that we rarely train them to do that.

I think that build up to tensor fields should be in every school program. If you can’t think of a field, you’re mathematically disabled and too many basic ideas about real world are inaccessible to you. This limits the ability to vote on a set of topics and participate in non-local decisions that involve systemic understanding. Same for formal logic and statistics.

Once familiarized with that, you can easily start thinking of nonlinearly signed areas, complex areas and areas simultaneously positive and negative by an attribute.