Learn Russian: Он очень упорный человек.

You are right to notice that! While both can translate to "stubborn," упрямый has a negative connotation (obstinate, refusing to listen). Упорный usually has a positive or neutral connotation, meaning persistent, tenacious, or determined. Describing someone as an упорный человек means they don't give up easily, though it can cross into "stubborn" depending on context.

How could we connect this sentence to the lesson's theme using всё-таки?

You can use всё-таки (nevertheless / still) to introduce a surprising contrast. For example: Он очень упорный человек, но всё-таки он согласился (He is a very stubborn person, but nevertheless he agreed). It highlights that his agreement happened despite his usual tenacious nature.

Where does the word упорный come from?

It shares a root with the noun упор (a support or prop) and the verb упираться (to brace oneself, to dig one's heels in). It literally describes someone who has planted their feet firmly and refuses to be pushed back.

Could I drop the word человек and just say Он очень упорный?

Yes, absolutely. Just like in English, you can use the adjective on its own as the predicate. Adding человек makes it slightly more emphatic about his fundamental character (he is that kind of person), but both versions are completely natural and common.