I think it has to do with how ridiculously and stereotypically American the character of Armstrong is.

On political ideology, he's essentially an Ancap, but what makes him so memorable is his attitude, mannerism, the lines, the voice actor. He's a walking meme that deserved more screen time.
He was hyped up previously, and when he arrives he steals the show completely from all the other characters.

This stands out in a Metal Gear game because everyone gets his share of lines, backstory, motives and connections, while instead Armstrong just came out of nowhere, roided up, and thew his character onto us one punch at the time. And we all loved it

What I really like about him is the right amount of comedy and seriousness,which many villains fail to accomplish to keep both. The DLC also gives us more of him and Monsoon and me appreciate them even more.
All made even better with both fantastic boss fights and theme songs.