The attacks I'm aware of use processor timings to measure the effects their programs are having. However, because an asynchronous circuit doesn't complete tasks

The attacks I'm aware of use processor timings to measure the effects their programs are having. However, because an asynchronous circuit doesn't complete tasks in a reliable cycle, you can't measure how the pipeline is being effected by your program in the same way. You could find new ways to force the processor to act in a reliable manner that you could measure, but that may change quite randomly from processor to processor, or even from moment to moment, depending on external environmental factors.