Uh, no...They were created via the "Power Spots" found in El Nido (which would mean, what? they're some byproduct of Chronopolis creating El Nido?)...And yeah, just because they can doesn't mean they would...but in this instance you might as well say they will if you're going to go off of a specific equipment set-up for their battle against one another...
And, also, would Serge still have the Chrono Cross anyways...? I don't think he would...But since this is a ridiculous scenario anyways, maybe you're (MDenham) thinking that it's taking place in that small window when Serge got the Chrono Cross and then used it on the Devourer of Time? I suppose it's debatable what happened afterwords...probably depending on where you stand in regards to the merge or something...
I still say Crono did more and is more skilled (not limited to just what he did, but certainly a majority of it).
Defeating Lavos>Playing a song at the Devourer of Time