To begin with I would just keep on oiling. Make sure you detect all joints and hinges, it can take a bit of repeated applications to get oil too reach the inners of moving parts too. Some treadle irons even have ball bearings, others are more hinged together. I swear Finish Line Ceramic Wet Lube has some advantages, it's very light but still coats and last quite a bit during use. I've used it on gear hubs as well as sewing machines for years and it has not caused any problems. Some treadles were meant to be oiled all the way, others have a few points you can use grease. I guess TriFlow will be extra smooth since it has teflon.
The problem with alternative oils are to find one with out too many addtives not suitable for the purpose. For grease you need to take it all appart and clean it to get to all the parts. Oil will seep in everywhere with a bit of effort and repeated application. Ball bearings need to be replaced some times, especially if they have been run with out lubrication. For treadle irons, it relatively easy to take them appart and clean them, so if you accept experimenting and an extra clean up job, you can try other oils. There's lots out there, I have not tried all, but besides ordinary sewing machine oil, and pure mineral oil, the above mentioned oils are the only ones I dare recommend for the entire machine. The advantage with grease is mostly less frequent application.
As leonf mentions, both machine and treadle parts need to run smoothly and effortlessly.