I find the publication order more comforting. You meet the most beloved characters first. Royce and Hadrian are your gateway into this world. When you later read about the ancient heroes they occasionally mention, it feels like uncovering treasured family history. Starting with the ancient history feels more distant, less personal.
They fit as evidence of author-reader synergy. Sullivan wrote the Riyria Revelations as a complete tale. Reader demand for more Royce and Hadrian led to the Chronicles. That's why they feel so lovingly crafted—they're born from genuine affection for the characters, both from the author and the fans. In the reading order, they represent the 'extra content' that exists because the core story was so good it left people wanting more. That's a pretty great place to fit.
The purest, most enjoyable experience is the publication order of the core Riyria books: 'The Riyria Revelations' ('Theft of Swords', etc.) first, THEN go back and read 'The Riyria Chronicles'. Those chronicle books are like delicious dessert—extra adventures with characters you already love. Reading them first would be like eating your cake before the main course; it's still sweet, but the meal's structure is off. Save the prequel treats for after you're fully invested in the duo.