Long story short: No because the Olympic Games in classical Greece was the most famous Panhellenic game and as such was thought of as the most glorifying. In the revival, a Greek poet called Soutsos urged for the glorification of mid-19th century Greece through a revival of the Olympic Games and that specific Panhellenic game stuck with future revivalists.

I am summarizing this quite a bit from a book called "A brief History of the Olympic Games" by David C. Young but the revival of the games started by a Greek poet Panagiotis Soutsos who wrote a poem called "Dialogue of the Dead" in 1833 in which during a conversation with Plato's ghost, the ghost asks "Where are your Olympic games?" as a way to question why 19th century Greece is in shambles compared to the glory of classical greece. Young points out why Soutsos used the Olympic games and not one of the other Panhellenic games for the revival of Greek glory: "In antiquity the Olympic Games symbolized excellence and prestige, a focal point for all of Greek culture" (Young 141). This is important because if Soutsos has called for a different Panhellenic Game then we might have a different name for our modern Olympic Game.

What we, as historians, classify as modern Olympic Games did not start until a Frenchmen named Pierre de Coubertin came along. Even then, the Olympic revival did not become a smashing success until the 1908 London Games as the 1896,1900, and 1904 games were largely problematic due to location or political agendas.

The closest we get to a Panhellenic Game by Coubertin is the 1906 Intercalated games which were supposed to be the Olympic Games held only in Athens, Greece every two years between the Olympic Games (which were held internationally). You can read about these games in journal by Karl Lennartz called "The 2nd International Olympic Games in Athens 1906". This game, although sharing the same 2-year post olympic setting as the Panhellenic "Pythian Games" had, was created with no thought of the Pythian Games in mind (hence the different name) and as such is not considered a revival of a different Panhellenic game.

The Olympic Game revival began as a platform to get schoolchildren into athletics at school because there were a lot of philosophers in the late 19th century suggesting the training of both the mind and the body and through various social-Darwinistic viewpoints, that the English population was not ready for war because they allow the weak to breed. So the Olympic Games were a way to ensure school children would be fit so they could get glory in the Olympic games. This is evident in the W.P. Brookes' revival of the Olympic games in Wenlock, England. They were eventually called the Wenlock Games and were fundamental to Coubertin's influence in a modern revival. For more on this read "Olympism" by Pierre de Coubertin and edited by Norbert Muller.

Sources:

"Dialogue of the Dead" by Panagiotis Soutsos

"A Brief History of the Olympic Games" by David C. Young

"The 2nd International Olympic Games in Athens 1906" by Karl Lennartz

"Olympism" by Pierre de Coubertin and edited by Norbert Muller