I believe the reason you don't see much growing after burning is that the seeds from weeds and some grasses may be burned by the fire since they are also on top of the ground. Burning wipes out weeds while allowing fresh grass growth from established roots below ground. I think burning will produce very little effect on your asparagus crowns that should be 4"-6" deep in the ground. People even spread salt lightly on the surface to kill weeds and inhibit germination of noxious plant seeds. I see burned areas in the medians of highways that rapidly sprout grasses and look better than any surrounding unburned area. Even forests get prescribed burns to kill off undergrowth and encourage new growth of grasses and tree seedlings.
Also, asparagus beds are very tolerant of light salt treatment. The salt treatment will kill off any seedlings, though. What you may find with the asparagus, if burned, is that you have very few seedlings coming up "volunteer." I have thousands of asparagus plants coming up every year from the seeds that are broadcast over the ground. By letting them grow and adding compost to the beds each year, I assure myself of ever-increasing asparagus beds. I really think your best solution will be to shred your beds using your rotary cutter or mower so that the seeds and tops are spread around over the area and decompose rapidly and naturally. The mature canes will lay there for years without decomposing.
If you want, you can line your fence with the canes to discourage weed growth due to decreased sunlight to the ground in those areas. If I was going to burn the tops, I think I'd do it off to the side in a pile and then spread the ash back over the beds. As you cut the tops, the mature seeds will fall and you won't kill them with your fire. Spreading the ash back over the beds will then not kill off the tiny volunteer seedlings. I just think burning over the beds won't hurt a thing except for seedlings and noxious weeds that have seeds laying on top of the ground. The asparagus crowns and roots should be well below any heat-sensitive zone.