You certainly get organisms that produce only short stretches of SC where the obligate CO is located, whilst the rest of the chromosome remains unsynapsed (eg Fritillaria and grashopper). Whether there are higher organisms with no SC that maintain the obligate CO, I don't know.
I don't think the S. Pombe approach would be reliable enough for higher organisms in general, but there are lots of weird and wonderful creatures that we know nothing about, so something similar could exist in nature. S. pombe has the 'linear elements' but not the central element, so is not so diverged. However, as it lacks the class I pathway of COs, this would be a disadvantage to most eukaryotes.